Transformation of spherical harmonics under rotation is a major problem in many areas of theoretical and applied science. While elegantly and efficiently solved for complex spherical harmonics with Wigner D- and d-matrices, extending this method to real spherical harmonics (RSH) faces serious difficulties not yet overcome. This work presents novel explicit formulas and recurrence relations for building RSH rotation matrices with lesser complexity and better computational efficiency. It also gives general closed forms of Wigner d-matrix elements in terms of the rotation angle instead of half this angle as is usual.

By the end of the eighteenth century, an important concern for mathematicians and physicists was the attraction of spheroids and the movement of the planets. In a 1782 memoir, Pierre-Simon de Laplace introduced a function V, later called “potential” by George Green (1828), which obeys the partial differential equation of which he is the eponym. He then took benefit from two memoirs (1783) of Adrien-Marie Legendre and, in the following year, presented a solution by a series expansion in terms of functions which were first referred to as Laplace's coefficients and later as associated Legendre functions. Eduard Heine called them “Kugelfunctionen”, that is “spherical functions”, in his 1861 “Handbuch der Kugelfunctionen” and the term “spherical harmonic analysis” was coined in 1867 by Sir William Thomson and Peter Guthrie Tait in their “Treatise on natural philosophy”. In 1877, Norman Macleod Ferrers wrote the first book entirely devoted to spherical harmonics (SH), “An elementary treatise on spherical harmonics and subjects connected with them”, not as elementary as the title suggests. Physics (potential theory) and geophysics (Earth's gravitational and magnetic fields) then made extensive use of real spherical harmonics (RSH) together with other so-called “special functions”.

The advent of quantum mechanics opened a new workspace for SH with J. C. Slater's seminal paper1 and the book by E. U. Condon and G. H. Shortley,2 which has become a reference. Group theory brought additional developments with contributions especially from G. Racah and E. P. Wigner, including the famous 1940 unpublished paper by the latter3 following his 1931 book written in German and translated to English in 1959.4 All these developments involve complex spherical harmonics (CSH) and the now called Wigner D- and d-matrices for dealing with their rotations.

Computational chemistry makes extensive use of rotations for both CSH and RSH with methods derived from Wigner d-matrix formalism.5,6 The latter paper is a key reference in the field. Its introduction clearly outlines the importance of RSH in molecular calculations and the need for a recursive approach when higher degrees of harmonics are required. However, even if the authors claim that “the described recursion scheme offers an eminently practical route combining simplicity, efficiency, and generality”, implementing it in a computer code is certainly a tough task.

More recently, SH showed up in the field of computer graphics7,8 where algorithms for fast real-time rotations of RSH are needed and again, to our knowledge, all methods are based on the aforementioned formalism. In reference 8, comparisons between different algorithms are reported with a particular mention of paper 6if you can fight your way through the math.

The purpose of this work is to introduce novel explicit formulas and recurrence relations for building RSH rotation matrices with lesser complexity and better computational efficiency.In the light of previous work, a first look to formulas (55) and (65) should support this statement of simplicity and efficiency.

1. Definition

The most concise definitions of Legendre polynomials Pn(x) and of associated Legendre functions |$P_n^m (x)$|Pnm(x) were given by Benjamin-Olinde Rodrigues in his thesis (Paris, France, 1815), definitions taken up by Ferrers in his aforementioned treatise and now known as Rodrigues’ formulas:

\begin{equation} \begin{array}{l} P_n (x) = \displaystyle\frac{1}{{2^n n!}}\displaystyle\frac{{d^n }}{{dx^n }}\left( {x^2 - 1} \right)^n, \\[10pt] P_n^m (x) = \displaystyle\frac{1}{{2^n n!}}\left( {1 - x^2 } \right)^{\frac{m}{2}} \displaystyle\frac{{d^{n + m} }}{{dx^{n + m} }}\left( {x^2 - 1} \right)^n = \left( {1 - x^2 } \right)^{\frac{m}{2}} \displaystyle\frac{{d^m }}{{dx^m }}P_n (x), \\[10pt] - 1 \le x \le 1,0 \le m \le n, \end{array} \end{equation}
Pn(x)=12nn!dndxnx21n,Pnm(x)=12nn!1x2m2dn+mdxn+mx21n=1x2m2dmdxmPn(x),1x1,0mn,
(1)

where n is the degree and m the order. A great variety of explicit formulas can be found in classical references 9–11 and can be rewritten in different ways, like the next two which involve only binomial coefficients:

\begin{equation} \hspace*{-9pt}\begin{array}{l} P_n^m (\cos \vartheta) = \displaystyle\frac{1}{{2^m m!}}\displaystyle\frac{{\left( {n + m} \right)!}}{{\left( {n - m} \right)!}}\sin ^m \vartheta \displaystyle\sum\limits_{p = 0}^{n - m} {\left( { - 1} \right)^p \displaystyle\frac{{\left( {\begin{array}{*{20}c} {n - m} \\[10pt] p \\ \end{array}} \right)\left( {\begin{array}{*{20}c} {n + m + p} \\[10pt] p \end{array}} \right)}}{{\left( {\begin{array}{*{20}c} {m + p} \\[10pt] p \end{array}} \right)}}\left( {\displaystyle\frac{{1 - \cos \vartheta }}{2}} \right)^p }, \\[10pt] P_n (\cos \vartheta) = \displaystyle\sum\limits_{p = 0}^n {\left( { - 1} \right)^p \left( {\begin{array}{*{20}c} n \\[10pt] p \end{array}} \right)\left( {\begin{array}{*{20}c} {n + p} \\[10pt] p \end{array}} \right)\left( {\displaystyle\frac{{1 - \cos \vartheta }}{2}} \right)^p }. \\ \end{array} \end{equation}
Pnm(cosϑ)=12mm!n+m!nm!sinmϑp=0nm1pnmpn+m+ppm+pp1cosϑ2p,Pn(cosϑ)=p=0n1pnpn+pp1cosϑ2p.
(2)

For numerical evaluations, recurrence relations must definitely be preferred to such formulas. However, not all the possible recursions are computationally stable and the following should be retained:12 

\begin{equation} \begin{array}{l} P_n^n = \left( {2n - 1} \right)!!\,\sin ^n \vartheta, \\[10pt] P_{n + 1}^n = \left( {2n + 1} \right)\cos \vartheta P_n^n, \\[10pt] \left( {n - m} \right)P_n^m = \left( {2n - 1} \right)\cos \vartheta P_{n - 1}^m - \left( {n + m - 1} \right)P_{n - 2}^m. \end{array} \end{equation}
Pnn=2n1!!sinnϑ,Pn+1n=2n+1cosϑPnn,nmPnm=2n1cosϑPn1mn+m1Pn2m.
(3)

N.B.: The double factorial notation9 stands for

\begin{equation*} \left( {2n - 1} \right)!! = 1 \cdot 3 \cdot 5 \ldots \left( {2n - 1} \right) = \displaystyle\frac{{2^n }}{{\sqrt \pi }}\Gamma \left( {n + \displaystyle\frac{1}{2}} \right)\,{\rm or}\,\left( {2n} \right)!! = 2 \cdot 4 \ldots \left( {2n} \right) = 2^n n!. \end{equation*}
2n1!!=1·3·5...2n1=2nπΓn+12 or 2n!!=2·4...2n=2nn!.

The above formulas for the associated Legendre functions do not contain a ( − 1)m factor that can be found elsewhere and creates some confusion in the scientific literature. The reason for this will be given below (Section II A 2) when evoking the Condon–Shortley phase factor.

2. Addition theorems

The so-called “addition theorem” of Legendre polynomials10 is very important for solving potential problems (expansion of the Green function |$1/\left| {\overrightarrow {r_2 } - \overrightarrow {r_1 } } \right|$|1/r2r1 in spherical coordinates) and will be very useful for the present work:

\begin{equation} \begin{array}{l} P_n \left( {\cos \vartheta _1 \cos \vartheta _2 + \sin \vartheta _1 \sin \vartheta _2 \cos \psi } \right) = P_n (\cos \vartheta _1)P_n (\cos \vartheta _2) \\[10pt] + 2\displaystyle\sum\limits_{m = 1}^n {\displaystyle\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}P_n^m (\cos \vartheta _1)P_n^m (\cos \vartheta _2)\cos m\psi },\quad 0 \le \vartheta _1 \le \pi,\quad 0 \le \vartheta _2 \le \pi. \\ \end{array} \end{equation}
Pncosϑ1cosϑ2+sinϑ1sinϑ2cosψ=Pn(cosϑ1)Pn(cosϑ2)+2m=1nnm!n+m!Pnm(cosϑ1)Pnm(cosϑ2)cosmψ,0ϑ1π,0ϑ2π.
(4)

Its equivalent for associated Legendre functions is less well known. As Legendre polynomials, these functions are a special case of Gegenbauer (also called ultraspherical) polynomials |$C_n^\lambda (x)$|Cnλ(x) which are the coefficients of αn in the following power-series expansion:

\begin{equation} \begin{array}{l} \left( {1 - 2x\alpha + \alpha ^2 } \right)^{ - \lambda } = \displaystyle\sum\limits_{n = 0}^\infty {C_n^\lambda (x)\,\alpha ^n }, \\[10pt] C_n^{\frac{1}{2}} (t) = P_n (t), \\[10pt] C_{n - m}^{m + \frac{1}{2}} (x) = \displaystyle\frac{{\left( {1 - x^2 } \right)^{ - \frac{m}{2}} }}{{\left( {2m - 1} \right)!!}}P_n^m (x). \\ \end{array} \end{equation}
12xα+α2λ=n=0Cnλ(x)αn,Cn12(t)=Pn(t),Cnmm+12(x)=1x2m22m1!!Pnm(x).
(5)

Their addition theorem can be found in references 10 and 11 and is rewritten in terms of associated Legendre functions (m ⩾ 1) as

\begin{equation} \begin{array}{l} \cos \Theta = \cos \vartheta _1 \cos \vartheta _2 + \sin \vartheta _1 \sin \vartheta _2 \cos \psi \\[10pt] P_n^m \left( {\cos \Theta } \right) = 2^m \left( {m - 1} \right)!\left( {\displaystyle\frac{{\sin \Theta }}{{\sin \vartheta _1 \sin \vartheta _2 }}} \right)^m \\[10pt] \times \displaystyle\sum\limits_{k = 0}^{n - m} {\left( {m + k} \right)\displaystyle\frac{{\left( {n - m - k} \right)!}}{{\left( {n + m + k} \right)!}}P_n^{m + k} (\cos \vartheta _1)P_n^{m + k} (\cos \vartheta _2)C_k^m (\cos \psi)} \\[10pt] = 2^m \left( {m - 1} \right)!\left( {\displaystyle\frac{{\sin \Theta }}{{\sin \vartheta _1 \sin \vartheta _2 }}} \right)^m \displaystyle\sum\limits_{k = m}^n {k\displaystyle\frac{{\left( {n - k} \right)!}}{{\left( {n + k} \right)!}}P_n^k (\cos \vartheta _1)P_n^k (\cos \vartheta _2)C_{k - m}^m (\cos \psi)}, \\[10pt] C_{k - m}^m (\cos \psi) = \displaystyle\sum\limits_{p = 0}^{k - m} {\displaystyle\frac{{\left( {m + p - 1} \right)!\left( {k - p - 1} \right)!}}{{p!\left( {k - m - p} \right)!\left[ {\left( {m - 1} \right)!} \right]^2 }}\cos \left[ {\left( {k - m - 2p} \right)\psi } \right]}. \\ \end{array} \end{equation}
cosΘ=cosϑ1cosϑ2+sinϑ1sinϑ2cosψPnmcosΘ=2mm1!sinΘsinϑ1sinϑ2m×k=0nmm+knmk!n+m+k!Pnm+k(cosϑ1)Pnm+k(cosϑ2)Ckm(cosψ)=2mm1!sinΘsinϑ1sinϑ2mk=mnknk!n+k!Pnk(cosϑ1)Pnk(cosϑ2)Ckmm(cosψ),Ckmm(cosψ)=p=0kmm+p1!kp1!p!kmp!m1!2coskm2pψ.
(6)

1. Definition

Following reference 2, CSH are defined as:

\begin{equation} \begin{array}{l} Y_n^m (\vartheta,\varphi) = \left( { - 1} \right)^m N_n^m P_n^m (\cos \vartheta)\exp \left( {im\varphi } \right), \\[10pt] Y_n^{ - m} (\vartheta,\varphi) = \left( { - 1} \right)^m \left[ {Y_n^m (\vartheta,\varphi)} \right]^ *, \\[10pt] N_n^m = \sqrt {\displaystyle\frac{{2n + 1}}{{4\pi }}\displaystyle\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}}. \\ \end{array} \end{equation}
Ynm(ϑ,φ)=1mNnmPnm(cosϑ)expimφ,Ynm(ϑ,φ)=1mYnm(ϑ,φ)*,Nnm=2n+14πnm!n+m!.
(7)

The normalization coefficient is chosen in order that the CSH are an orthonormal basis for square-integrable functions on the unit sphere:

\begin{equation} \int_0^{2\pi }\!\!\! {\int_0^\pi {Y_n^m (\vartheta,\varphi)\left[ {Y_{n'}^{m'} (\vartheta,\varphi)} \right]^ * \sin \vartheta \,d\vartheta d\varphi } } = \delta _n^{n'} \delta _m^{m'} \end{equation}
02π0πYnm(ϑ,φ)Ynm(ϑ,φ)*sinϑdϑdφ=δnnδmm
(8)

2. The Condon–Shortley phase factor

In order to understand the genesis of the above-mentioned confusion (Section II A 1), one must read §4 “Orbital angular momentum” of their book.2 They first find the solution numbered (10) for m = n, written in their notations (changing only l in n and ml in m)

\begin{equation} \Theta (n,n) = \left( { - 1} \right)^n \sqrt {\displaystyle\frac{{\left( {2n + 1} \right)!}}{2}} \displaystyle\frac{1}{{2^n n!}}\sin ^n \vartheta, \end{equation}
Θ(n,n)=1n2n+1!212nn!sinnϑ,
(9)

with the comment: “The phase has been taken as ( − 1)nfor convenience in later work”. The demonstration then leads to the results numbered (17) and (18)

\begin{eqnarray} \Theta (n,m) &=& \left( { - 1} \right)^m \sqrt {\displaystyle\frac{{2n + 1}}{2}\displaystyle\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}} \sin ^m \vartheta \displaystyle\frac{{d^m }}{{\left( {d\cos \vartheta } \right)^m }}P_n (\cos \vartheta), \nonumber\\ \Theta (n, - m) &=& + \sqrt {\displaystyle\frac{{2n + 1}}{2}\displaystyle\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}} \sin ^m \vartheta \displaystyle\frac{{d^m }}{{\left( {d\cos \vartheta } \right)^m }}P_n (\cos \vartheta), \\ \Theta (n, - m) &=& \left( { - 1} \right)^m \Theta (n,m),\;m > 0, \nonumber \end{eqnarray}
Θ(n,m)=1m2n+12nm!n+m!sinmϑdmdcosϑmPn(cosϑ),Θ(n,m)=+2n+12nm!n+m!sinmϑdmdcosϑmPn(cosϑ),Θ(n,m)=1mΘ(n,m),m>0,
(10)

with the comment: “The natural choice of phases which we have made here leads to a rather curious occurrence of the factor −1 only for positive odd values of m. If we had approached the problem through the usual form of the theory of spherical harmonics the natural tendency would have been to choose the normalizing factors with omission of the ( − 1)min these formulas”. It is thus clear that they did not intend to include this factor in the Rodrigues’ formula which was stillthe case for some of the authors.

For our part, we shall leave this factor to the CSH and keep the associated Legendre functions in their original form which gives n = m = 1:

\begin{equation} \begin{array}{l} Y_1^1 (\vartheta,\varphi) = - \sqrt {\displaystyle\frac{3}{{8\pi }}} \sin \vartheta \exp \left( {i\varphi } \right), \\[10pt] P_1^1 (\cos \vartheta) = + \sin \vartheta. \\ \end{array} \end{equation}
Y11(ϑ,φ)=38πsinϑexpiφ,P11(cosϑ)=+sinϑ.
(11)

1. Definition

The most straightforward way to define RSH is:

\begin{equation} \begin{array}{l} E_n^m \left( {\vartheta,\varphi } \right) = P_n^m (\cos \vartheta)\cos m\varphi ,\;m = 0,1,2, \ldots,n, \\[10pt] O_n^m \left( {\vartheta,\varphi } \right) = P_n^m (\cos \vartheta)\sin m\varphi ,\;m = 1,2, \ldots,n, \\ \end{array} \end{equation}
Enmϑ,φ=Pnm(cosϑ)cosmφ,m=0,1,2,...,n,Onmϑ,φ=Pnm(cosϑ)sinmφ,m=1,2,...,n,
(12)

building a set of 2n + 1 functions for each degree n, naturally the same number as the CSH for the same degree. These definitions keep only the bare variable parts of the functions without any weighting or normalizing coefficients which will be discussed later.

We shall call the |$E_n^m $|Enm “even” and the |$O_n^m $|Onm “odd” RSH within the meaning of the Fourier series in φ (not to be confused with parity of CSH).

These functions are orthogonal under integration over the surface of the unit sphere:

\begin{equation} \begin{array}{l} \displaystyle\frac{{2n + 1}}{{4\pi }}\int_0^{2\pi } {\int_0^\pi {E_n^m \left( {\vartheta,\varphi } \right)E_\nu ^\mu \left( {\vartheta,\varphi } \right)\sin \vartheta d\vartheta d\varphi } } = \displaystyle\frac{{1 + \delta _m^0 }}{2}\displaystyle\frac{{\left( {n + m} \right)!}}{{\left( {n - m} \right)!}}\delta _n^\nu \delta _m^\mu, \\[10pt] \displaystyle\frac{{2n + 1}}{{4\pi }}\int_0^{2\pi } {\int_0^\pi {O_n^m \left( {\vartheta,\varphi } \right)O_\nu ^\mu \left( {\vartheta,\varphi } \right)\sin \vartheta d\vartheta d\varphi } } = \displaystyle\frac{1}{2}\displaystyle\frac{{\left( {n + m} \right)!}}{{\left( {n - m} \right)!}}\delta _n^\nu \delta _m^\mu, \\[10pt] \displaystyle\frac{{2n + 1}}{{4\pi }}\int_0^{2\pi } {\int_0^\pi {E_n^m \left( {\vartheta,\varphi } \right)O_\nu ^\mu \left( {\vartheta,\varphi } \right)\sin \vartheta d\vartheta d\varphi } } = 0. \\ \end{array} \end{equation}
2n+14π02π0πEnmϑ,φEνμϑ,φsinϑdϑdφ=1+δm02n+m!nm!δnνδmμ,2n+14π02π0πOnmϑ,φOνμϑ,φsinϑdϑdφ=12n+m!nm!δnνδmμ,2n+14π02π0πEnmϑ,φOνμϑ,φsinϑdϑdφ=0.
(13)

2. Weighting vs. normalization

From a mathematical viewpoint, there is no need for extra coefficients to transform the above defined functions under a rotation of the reference frame. As already mentioned,this is not the case for numerical evaluations for the stability of recurrences. Indeed, if the range of the Legendre polynomials Pn(cos ϑ) is strictly [ − 1, +1], that of the associated Legendre functions |$P_n^m (\cos \vartheta)$|Pnm(cosϑ) increases very fast with their degree and order as a consequence of the factorials in relation (3). The normalization coefficient |$N_n^m $|Nnm of relation (7) overcomes this drawback for the CSH and could also be used for the RSH but a more elegant solution has been proposed by A. Schmidt.13 It is known in the geophysics community as the Schmidt quasi-normalization, replacing the original Legendre functions by the following:

\begin{equation} P_{Sn}^m (\cos \vartheta) = \sqrt {\left( {2 - \delta _m^0 } \right)\displaystyle\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}} \;P_n^m (\cos \vartheta) ,\;0 \le m \le n. \end{equation}
PSnm(cosϑ)=2δm0nm!n+m!Pnm(cosϑ),0mn.
(14)

One recognizes a part of the CSH normalization coefficient but this definition leaves unchanged the Legendre polynomials (m = 0) and, as we shall see later, brings important simplifications in several formulas. As an example, already emphasized by Schmidt, the “addition theorem” of Legendre polynomials takes a much simpler form than the original (4):

\begin{equation} P_{Sn}^0 \left( {\cos \vartheta _1 \cos \vartheta _2 + \sin \vartheta _1 \sin \vartheta _2 \cos \psi } \right) = \displaystyle\sum\limits_{m = 0}^n {P_{Sn}^m (\cos \vartheta _1)P_{Sn}^m (\cos \vartheta _2)\cos m\psi }. \end{equation}
PSn0cosϑ1cosϑ2+sinϑ1sinϑ2cosψ=m=0nPSnm(cosϑ1)PSnm(cosϑ2)cosmψ.
(15)

The orthogonality relations (13) also become simpler:

\begin{equation} \begin{array}{l} \displaystyle\frac{{2n + 1}}{{4\pi }}\int_0^{2\pi } {\int_0^\pi {E_{Sn}^m \left( {\vartheta,\varphi } \right)E_{S\nu }^\mu \left( {\vartheta,\varphi } \right)\sin \vartheta d\vartheta d\varphi } } = \delta _n^\nu \delta _m^\mu, \\[10pt] \displaystyle\frac{{2n + 1}}{{4\pi }}\int_0^{2\pi } {\int_0^\pi {O_{Sn}^m \left( {\vartheta,\varphi } \right)O_{S\nu }^\mu \left( {\vartheta,\varphi } \right)\sin \vartheta d\vartheta d\varphi } } = \delta _n^\nu \delta _m^\mu, \\[10pt] \displaystyle\frac{{2n + 1}}{{4\pi }}\int_0^{2\pi } {\int_0^\pi {E_{Sn}^m \left( {\vartheta,\varphi } \right)O_{S\nu }^\mu \left( {\vartheta,\varphi } \right)\sin \vartheta d\vartheta d\varphi } } = 0. \\ \end{array} \end{equation}
2n+14π02π0πESnmϑ,φESνμϑ,φsinϑdϑdφ=δnνδmμ,2n+14π02π0πOSnmϑ,φOSνμϑ,φsinϑdϑdφ=δnνδmμ,2n+14π02π0πESnmϑ,φOSνμϑ,φsinϑdϑdφ=0.
(16)

The numerically stable recurrences given in (3) become:

\begin{equation} \begin{array}{l} P_{Sn}^n = \sqrt {2\displaystyle\frac{{\left( {2n - 1} \right)!!}}{{\left( {2n} \right)!!}}} \sin ^n \vartheta, \\[10pt] P_{Sn + 1}^n = \sqrt {2n + 1} \cos \vartheta P_{Sn}^n, \\[10pt] \sqrt {\left( {n - m} \right)\left( {n + m} \right)} P_{Sn}^m = \left( {2n - 1} \right)\cos \vartheta P_{Sn - 1}^m - \sqrt {\left( {n - m - 1} \right)\left( {n + m - 1} \right)} P_{Sn - 2}^m. \\ \end{array} \end{equation}
PSnn=22n1!!2n!!sinnϑ,PSn+1n=2n+1cosϑPSnn,nmn+mPSnm=2n1cosϑPSn1mnm1n+m1PSn2m.
(17)

They can be coded as easily as the original ones but square roots make the mathematical work more intricate. For this reason, most of our demonstrations will be carried out by using the RSH as defined in equations (12) and their results will then be transformed by introducing the quasi-normalized functions.

N.B.: Another “normalization” by a factor |$\displaystyle\frac{{\left( {n - m} \right)!}}{{\left( {2n - 1} \right)!!}}$|nm!2n1!! is also known in geophysics as “Gauss normalization”.

In the context of magnetism, especially for describing the magnetic field of NMR or MRI magnets, each Cartesian component of the field in an empty region of space obeys the Laplace equation and thus can be expanded on the basis of regular or irregular solid SH. The main component of the field in the z direction in the central region of the magnet has the unique form

\begin{equation} \displaystyle\frac{{B_z }}{{B_0 }} = 1 + \displaystyle\sum\limits_{n = 1}^\infty {\left( {\displaystyle\frac{r}{{r_0 }}} \right)^n \left[ {Z_n P_n (\cos \vartheta) + \displaystyle\sum\limits_{m = 1}^n {\left( {X_n^m \cos m\varphi + Y_n^m \sin m\varphi } \right)W_n^m P_n^m (\cos \vartheta)} } \right]}, \end{equation}
BzB0=1+n=1rr0nZnPn(cosϑ)+m=1nXnmcosmφ+YnmsinmφWnmPnm(cosϑ),
(18)

which is valid inside the biggest “magnetically” empty sphere centered at the origin of the coordinate system. B0 is the field at the center and the coefficients |$Z_n,\;X_n^m,\;Y_n^m $|Zn,Xnm,Ynm fully describe the so-called inhomogeneity of the field. In order to give them a simple interpretation in relative value of the center field (usually in ppm for this type of magnet) on a reference sphere of radius r0, it is convenient to introduce a weighting factor |$W_n^m $|Wnm which would ideally make the range of |$| {W_n^m P_n^m (\cos \vartheta)} |$||WnmPnm(cosϑ)| to be [0, 1] as it is naturally for |$| {P_n^m (\cos \vartheta)} |$||Pnm(cosϑ)|. This ideal is out of the reach of simple calculations but the following heuristic expression gives an almost perfect result:

\begin{equation} W_n^m = \displaystyle\frac{{\left( {n - m - 1} \right)!!}}{{\left( {n + m - 1} \right)!!}} = \displaystyle\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}\displaystyle\frac{{\left( {n + m} \right)!!}}{{\left( {n - m} \right)!!}} \to P_{Wn}^m (\cos \vartheta) = W_n^m P_n^m (\cos \vartheta). \end{equation}
Wnm=nm1!!n+m1!!=nm!n+m!n+m!!nm!!PWnm(cosϑ)=WnmPnm(cosϑ).
(19)

This leaves the Legendre polynomialsunchanged, giving the ideal result for m = n, a range always smaller for mn but not too small, and it makes the transformation of equations much simpler than any normalization owing to the absence of square roots. For this last reason and to save space, the corresponding equations and tables will not be displayed in this paper.

A rotation which transforms the Cartesian reference frame Oxyz into Oxyz′ can be described by three successive rotations around three different specified axes and this can be done in various ways, including the aeronautic “Yaw-Pitch-Roll” convention. As chosen by Wigner,4 the successive rotation scheme is as follows:

  1. Oxyz is rotated by a first angle α around Oz → Ox1y1z;

  2. Ox1y1z is rotated by a second angle β around Oy1 → Ox2y1z2;

  3. Ox2y1z2 is rotated by a third angle γ around Oz2 → Ox3y3z2 which is relabeled as Oxyz′.

The three angles (α, β, γ) are called the Euler angles in the ZαYβZγ convention. Through elegant and powerful group symmetry arguments, Wigner has shown that the transformation of a CSH from one reference frame to the other can be expressed by means of the now called Wigner D-matrices, one from each degree n, the elements of which can be written as4 

\begin{equation} D_{m,m'}^n (\alpha,\beta,\gamma) = e^{im\alpha } d_{m,m'}^n (\beta)e^{im'\gamma } \end{equation}
Dm,mn(α,β,γ)=eimαdm,mn(β)eimγ
(20)

The transformations of CSH and RSH under Zα and Zγ rotations are trivial to produce since they affect only cos mφ and sin mφ terms in both reference frames. The Yβ rotation is much more difficult to handle since it affects simultaneously the two variables ϑ and φ. This is achieved through the now called Wigner d-matrices which are real square matrices, (2n + 1) × (2n + 1) for each degree n, the elements of which |$d_{m,m^\prime }^n (\beta)$|dm,mn(β) have been given explicit expressions by Wigner4 in terms of |$\cos \frac{\beta }{2}$|cosβ2 and |$\sin \frac{\beta }{2}$|sinβ2, not reproduced here (see Section III G) since they will not be used, contrary to all previous work on RSH rotations.

In his 1940 unpublished paper, page 118 of Ref. 3, Wigner points out that “the disadvantage of Eqs. […] is that they express the representation coefficients by means of the powers of cos ½ β and sin ½ β”. He then shows that the Yβ rotation can be transformed into successive rotations as

\begin{equation} Y_\beta = Z_{\pi /2} Y_{\pi /2} Z_\beta Y_{ - \pi /2} Z_{ - \pi /2} \end{equation}
Yβ=Zπ/2Yπ/2ZβYπ/2Zπ/2
(21)

The Y±π/2 can be calculated once and β enters only the trivialZβ. This trick is mentioned in Ref. 7 and discussed in Ref. 8 for its computational efficiency. Partly successful attempts have been made14 to get closed forms of d-matrix elements for |$\beta = {\raise0.5ex\hbox{\scriptstyle \pi }\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{\scriptstyle 2}}$|β=π/2, but this does not solve the problem of RSH rotations, which will be undertaken in Sections III F and III G.

We consider in the following a single rotation β around the Oy axis which changes Oxyz into Oxyz′ with Oy′ ≡ Oy. The spherical coordinates of a general point in these two reference frames are (r, ϑ, φ) and (r, ϑ′, φ′), respectively, linked by the following equations:

\begin{equation} \begin{array}{l} \sin \vartheta \cos \varphi = \sin \beta \cos \vartheta ' + \cos \beta \sin \vartheta '\cos \varphi ', \\[10pt] \sin \vartheta \sin \varphi = \sin \vartheta '\sin \varphi ', \\[10pt] \cos \vartheta = \cos \beta \cos \vartheta ' - \sin \beta \sin \vartheta '\cos \varphi '. \\ \end{array} \end{equation}
sinϑcosφ=sinβcosϑ+cosβsinϑcosφ,sinϑsinφ=sinϑsinφ,cosϑ=cosβcosϑsinβsinϑcosφ.
(22)

Nothing else will be needed to transform the RSH defined in Section II C 1.

As for Wigner d-matrices, the transformation formulas do not mix different degrees and are thus written as

\begin{equation} \begin{array}{l} E_n^m \left( {\vartheta,\varphi } \right) = \displaystyle\sum\limits_{k = 0}^n {C_n^{m,k} E_n^k \left( {\vartheta ',\varphi '} \right)},\quad \forall n \ge 0,\quad m \in \left[ {0,n} \right], \\[10pt] O_n^m \left( {\vartheta,\varphi } \right) = \displaystyle\sum\limits_{k = 1}^n {S_n^{m,k} O_n^k \left( {\vartheta ',\varphi '} \right)},\quad \forall n \ge 1,\quad m \in \left[ {1,n} \right]. \\ \end{array} \end{equation}
Enmϑ,φ=k=0nCnm,kEnkϑ,φ,n0,m0,n,Onmϑ,φ=k=1nSnm,kOnkϑ,φ,n1,m1,n.
(23)

At each degree n, the |$C_n^{m,k} $|Cnm,k build up a (n + 1) × (n + 1) square matrix Cn and the |$S_n^{m,k} $|Snm,k build up an n × n square matrix Sn (row index m, column index k), the elements of which are functions of β only. Besides the trivial C0 = [1], equations (22) give directly the degree 1 matrices:

\begin{equation} \begin{array}{l} C_1 = \left[ {\begin{array}{*{20}c} {\cos \beta } & { - \sin \beta } \\[7pt] {\sin \beta } & {\cos \beta } \\ \end{array}} \right] \\[13pt] S_1 = \left[ 1 \right] \\ \end{array} \end{equation}
C1=cosβsinβsinβcosβS1=1
(24)

Using the brute force of computer algebra software,such as Maple (not available at Wigner's time!), it is quite easy to create a program which generates the Cn and Sn matrices at any degree n from equations (12) and (22), through simple trigonometric transformations. These matrices can be stored for further use, notably for checking examples of the explicit formulas and the recurrence relations which will be established. This can also be done if the RSH are written with weighted or Schmidt quasi-normalized associated Legendre functions, in which cases the matrices would be noted CWn and SWn or CSn and SSn, respectively, with the following relations drawn from (14):

\begin{equation} \begin{array}{l} C_{Sn}^{m,k} = C_n^{m,k} \sqrt {\displaystyle\frac{{2 - \delta _m^0 }}{{2 - \delta _k^0 }}\displaystyle\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}\displaystyle\frac{{\left( {n + k} \right)!}}{{\left( {n - k} \right)!}}}, \\[18pt] S_{Sn}^{m,k} = S_n^{m,k} \sqrt {\displaystyle\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}\displaystyle\frac{{\left( {n + k} \right)!}}{{\left( {n - k} \right)!}}}. \\ \end{array} \end{equation}
CSnm,k=Cnm,k2δm02δk0nm!n+m!n+k!nk!,SSnm,k=Snm,knm!n+m!n+k!nk!.
(25)

At the first degree, CS1 = C1 and SS1 = S1. The second and third degree matrices are displayed below with c = cos β and s = sin β for short:

\begin{equation} C_2 = \left[ {\begin{array}{c@\quad c@\quad c} {\displaystyle\frac{3}{2}c^2 - \displaystyle\frac{1}{2}} & { - sc} & {\displaystyle\frac{1}{4}c^2 } \\[10pt] {3sc} & {2c^2 - 1} & { - \displaystyle\frac{1}{2}sc} \\[10pt] {3s^2 } & {2sc} & {\displaystyle\frac{1}{2}c^2 + \displaystyle\frac{1}{2}} \\ \end{array}} \right],S_2 = \left[ {\begin{array}{c@\quad c} c & { - \displaystyle\frac{1}{2}s} \\[10pt] {2s} & c \\ \end{array}} \right]. \end{equation}
C2=32c212sc14c23sc2c2112sc3s22sc12c2+12,S2=c12s2sc.
(26)
\begin{equation} C_{S2} = \left[ {\begin{array}{c@\quad c@\quad c} {\displaystyle\frac{3}{2}c^2 - \displaystyle\frac{1}{2}} & { - \sqrt 3 sc} & {\displaystyle\frac{{\sqrt 3 }}{2}s^2 } \\[10pt] {\sqrt 3 sc} & {2c^2 - 1} & { - sc} \\[10pt] {\displaystyle\frac{{\sqrt 3 }}{2}s^2 } & {sc} & {\displaystyle\frac{1}{2}c^2 + \displaystyle\frac{1}{2}} \\ \end{array}} \right] ,S_{S2} = \left[ {\begin{array}{c@\quad c} c & { - s} \\[10pt] s & c \\ \end{array}} \right]. \end{equation}
CS2=32c2123sc32s23sc2c21sc32s2sc12c2+12,SS2=cssc.
(27)
\begin{eqnarray} \begin{array}{l} C_3 = \left[ {\begin{array}{c@\quad c@\quad c@\quad c} {\displaystyle\frac{1}{2}c\left( {5c^2 - 3} \right)} & { - \displaystyle\frac{1}{4}s\left( {5c^2 - 1} \right)} & {\displaystyle\frac{1}{4}s^2 c} & { - \displaystyle\frac{1}{{24}}s^3 } \\[10pt] {\displaystyle\frac{3}{2}s\left( {5c^2 - 1} \right)} & {\displaystyle\frac{1}{4}c\left( {15c^2 - 11} \right)} & { - \displaystyle\frac{1}{4}s\left( {3c^2 - 1} \right)} & {\displaystyle\frac{1}{8}s^2 c} \\[10pt] {15s^2 c} & {\displaystyle\frac{5}{2}s\left( {3c^2 - 1} \right)} & {\displaystyle\frac{1}{2}c\left( {3c^2 - 1} \right)} & { - \displaystyle\frac{1}{4}s\left( {c^2 + 1} \right)} \\[10pt] {15s^3 } & {\displaystyle\frac{{15}}{2}s^2 c} & {\displaystyle\frac{3}{2}s\left( {c^2 + 1} \right)} & {\displaystyle\frac{1}{4}c\left( {c^2 + 3} \right)}\nonumber \\ \end{array}} \right], \\[32pt] S_3 = \left[ {\begin{array}{c@\quad c@\quad c} {\displaystyle\frac{5}{4}c^2 - \displaystyle\frac{1}{4}} & { - \displaystyle\frac{1}{2}sc} & {\displaystyle\frac{1}{8}s^2 } \\[10pt] {5sc} & {2c^2 - 1} & { - \displaystyle\frac{1}{2}sc} \\[10pt] {\displaystyle\frac{{15}}{2}s^2 } & {3sc} & {\displaystyle\frac{3}{4}c^2 + \displaystyle\frac{1}{4}} \\ \end{array}} \right]. \\[-1.3pc] \end{array} \\ \end{eqnarray}
C3=12c5c2314s5c2114s2c124s332s5c2114c15c21114s3c2118s2c15s2c52s3c2112c3c2114sc2+115s3152s2c32sc2+114cc2+3,S3=54c21412sc18s25sc2c2112sc152s23sc34c2+14.
(28)
\begin{eqnarray} \begin{array}{l} C_{S3} = \left[ {\begin{array}{c@\quad c@\quad c@\quad c} {\displaystyle\frac{1}{2}c\left( {5c^2 - 3} \right)} & { - \displaystyle\frac{{\sqrt 6 }}{4}s\left( {5c^2 - 1} \right)} & {\displaystyle\frac{{\sqrt {15} }}{2}s^2 c} & { - \displaystyle\frac{{\sqrt {10} }}{4}s^3 } \\[10pt] {\displaystyle\frac{{\sqrt 6 }}{4}s\left( {5c^2 - 1} \right)} & {\displaystyle\frac{1}{4}c\left( {15c^2 - 11} \right)} & { - \displaystyle\frac{{\sqrt {10} }}{4}s\left( {3c^2 - 1} \right)} & {\displaystyle\frac{{\sqrt {15} }}{4}s^2 c} \\[10pt] {\displaystyle\frac{{\sqrt {15} }}{2}s^2 c} & {\displaystyle\frac{{\sqrt {10} }}{4}s\left( {3c^2 - 1} \right)} & {\displaystyle\frac{1}{2}c\left( {3c^2 - 1} \right)} & { - \displaystyle\frac{{\sqrt 6 }}{4}s\left( {c^2 + 1} \right)} \\[10pt] {\displaystyle\frac{{\sqrt {10} }}{4}s^3 } & {\displaystyle\frac{{\sqrt {15} }}{4}s^2 c} & {\displaystyle\frac{{\sqrt 6 }}{4}s\left( {c^2 + 1} \right)} & {\displaystyle\frac{1}{4}c\left( {c^2 + 3} \right)} \\ \end{array}} \right] ,\nonumber \\[60pt] S_{S3} = \left[ {\begin{array}{c@\quad c@\quad c} {\displaystyle\frac{5}{4}c^2 - \displaystyle\frac{1}{4}} & { - \displaystyle\frac{{\sqrt {10} }}{2}sc} & {\displaystyle\frac{{\sqrt {15} }}{4}s^2 } \\[10pt] {\displaystyle\frac{{\sqrt {10} }}{2}sc} & {2c^2 - 1} & { - \displaystyle\frac{{\sqrt 6 }}{2}sc} \\[10pt] {\displaystyle\frac{{\sqrt {15} }}{4}s^2 } & {\displaystyle\frac{{\sqrt 6 }}{2}sc} & {\displaystyle\frac{3}{4}c^2 + \displaystyle\frac{1}{4}} \\ \end{array}} \right] . \\[-1.2pc] \end{array}\\ \end{eqnarray}
CS3=12c5c2364s5c21152s2c104s364s5c2114c15c211104s3c21154s2c152s2c104s3c2112c3c2164sc2+1104s3154s2c64sc2+114cc2+3,SS3=54c214102sc154s2102sc2c2162sc154s262sc34c2+14.
(29)

They exhibit remarkable symmetry properties. The proof for the CSn matrices is given below and it is a similar for those of the SSn:

\begin{equation} \begin{array}{l} P_{Sn}^m (\cos \vartheta)\cos m\varphi = \displaystyle\sum\limits_{k = 0}^n {C_{Sn}^{m,k} P_{Sn}^k (\cos \vartheta ')\cos k\varphi '}, \\[10pt] P_{Sn}^k (\cos \vartheta ')\cos k\varphi ' = \displaystyle\sum\limits_{\mu = 0}^n {\left( { - 1} \right)^{k + \mu } C_{Sn}^{k,\mu } P_{Sn}^\mu (\cos \vartheta)\cos \mu \varphi }, \\ \end{array} \end{equation}
PSnm(cosϑ)cosmφ=k=0nCSnm,kPSnk(cosϑ)coskφ,PSnk(cosϑ)coskφ=μ=0n1k+μCSnk,μPSnμ(cosϑ)cosμφ,
(30)

where the second relation comes directly from the inverse of relations (22). Using the orthogonality relations (16) one gets:

\begin{equation} \begin{array}{l} \displaystyle\frac{{2n + 1}}{{4\pi }}\int_0^{2\pi } {\int_0^\pi {P_{Sn}^m (\cos \vartheta)\cos m\varphi P_{Sn}^k (\cos \vartheta ')\cos k\varphi '\sin \vartheta 'd\vartheta 'd\varphi ' = C_{Sn}^{m,k} } }, \\[10pt] \displaystyle\frac{{2n + 1}}{{4\pi }}\int_0^{2\pi } {\int_0^\pi {P_{Sn}^k (\cos \vartheta ')\cos k\varphi 'P_{Sn}^m (\cos \vartheta)\cos m\varphi \sin \vartheta d\vartheta d\varphi = \left( { - 1} \right)^{k + m} C_{Sn}^{k,m} } }. \\ \end{array} \end{equation}
2n+14π02π0πPSnm(cosϑ)cosmφPSnk(cosϑ)coskφsinϑdϑdφ=CSnm,k,2n+14π02π0πPSnk(cosϑ)coskφPSnm(cosϑ)cosmφsinϑdϑdφ=1k+mCSnk,m.
(31)

Both integrals are equal since they are taken over the whole same unit sphere. It then follows:

\begin{equation} \begin{array}{l} C_{Sn}^{m,k} = \left( { - 1} \right)^{m + k} C_{Sn}^{k,m},\;\;0 \le m \le n,\;\;0 \le k \le n, \\[10pt] S_{Sn}^{m,k} = \left( { - 1} \right)^{m + k} S_{Sn}^{k,m},\;\;1 \le m \le n,\;\;1 \le k \le n. \\ \end{array} \end{equation}
CSnm,k=1m+kCSnk,m,0mn,0kn,SSnm,k=1m+kSSnk,m,1mn,1kn.
(32)

As a consequence, all these matrices are orthogonal with determinant unity.

From relations (25) one achieves the symmetry relations for the Cn and Sn matrices:

\begin{equation} \begin{array}{l} C_n^{0,m} = 2 \times \left( { - 1} \right)^m \displaystyle\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}C_n^{m,0},\;\;0 \le m \le n, \\[10pt] \displaystyle\frac{{C_n^{m,k} }}{{C_n^{k,m} }} = \displaystyle\frac{{S_n^{m,k} }}{{S_n^{k,m} }} = \left( { - 1} \right)^{m + k} \displaystyle\frac{{\left( {n + m} \right)!\left( {n - k} \right)!}}{{\left( {n - m} \right)!\left( {n + k} \right)!}},\;\;1 \le m \le n,\;\;1 \le k \le n . \\ \end{array} \end{equation}
Cn0,m=2×1mnm!n+m!Cnm,0,0mn,Cnm,kCnk,m=Snm,kSnk,m=1m+kn+m!nk!nm!n+k!,1mn,1kn.
(33)

These properties are much simpler for the RSH expressed in terms of Schmidt quasi-normalized functions and show up naturally as the sum of both a symmetric and an antisymmetric matrix which are obtained by replacing alternate diagonals by zeros, for instance:

\begin{equation} \begin{array}{l} C_{S2} = \left[ {\begin{array}{c@\quad c@\quad c} {\displaystyle\frac{3}{2}\cos ^2 \beta - \displaystyle\frac{1}{2}} & 0 & {\displaystyle\frac{1}{2}\sqrt 3 \sin ^2 \beta } \\[10pt] 0 & {2\cos ^2 \beta - 1} & 0 \\[10pt] {\displaystyle\frac{1}{2}\sqrt 3 \sin ^2 \beta } & 0 & {\displaystyle\frac{1}{2}\cos ^2 \beta + \displaystyle\frac{1}{2}} \\ \end{array}} \right] \\[40pt] + \left[ {\begin{array}{c@\quad c@\quad c} 0 & { - \sqrt 3 \sin \beta \cos \beta } & 0 \\[10pt] {\sqrt 3 \sin \beta \cos \beta } & 0 & { - \sin \beta \cos \beta } \\[10pt] 0 & {\sin \beta \cos \beta } & 0 \\ \end{array}} \right] . \\ \end{array} \end{equation}
CS2=32cos2β120123sin2β02cos2β10123sin2β012cos2β+12+03sinβcosβ03sinβcosβ0sinβcosβ0sinβcosβ0.
(34)

For β = 0, all matrices are identity, for β = π they are ±identity while for |$\beta = \pm \frac{\pi }{2}$|β=±π2, only either the symmetric or the antisymmetric matrix remains which is then orthogonal.

Rotating RSH instead of CSH seems to result in twice as much work but this is not true since there are crossed relations between the matrix elements of the even and odd RSH. As a consequence, only one type has to be calculated. These relations are obtained through a partial differentiation of definitions (12):

\begin{equation} \begin{array}{l} \left( {\displaystyle\frac{{\partial E_n^m }}{{\partial \varphi }}} \right)_\vartheta = - m{\kern 1pt} {\kern 1pt} O_n^m \\[15pt] = \displaystyle\sum\limits_{k = 0}^n {C_n^{m,k} \left[ {\displaystyle\frac{{dP_n^k (\cos \vartheta ')}}{{d\vartheta '}}\left( {\displaystyle\frac{{\partial \vartheta '}}{{\partial \varphi }}} \right)_\vartheta \cos k\varphi ' - k{\kern 1pt} P_n^k (\cos \vartheta ')\sin k\varphi '\left( {\displaystyle\frac{{\partial \varphi '}}{{\partial \varphi }}} \right)_\vartheta } \right]} , \\[15pt] \left( {\displaystyle\frac{{\partial O_n^m }}{{\partial \varphi }}} \right)_\vartheta = m{\kern 1pt} {\kern 1pt} E_n^m \\[15pt] = \displaystyle\sum\limits_{k = 1}^n {S_n^{m,k} \left[ {\displaystyle\frac{{dP_n^k (\cos \vartheta ')}}{{d\vartheta '}}\left( {\displaystyle\frac{{\partial \vartheta '}}{{\partial \varphi }}} \right)_\vartheta \sin k\varphi ' + k{\kern 1pt} P_n^k (\cos \vartheta ')\cos k\varphi '\left( {\displaystyle\frac{{\partial \varphi '}}{{\partial \varphi }}} \right)_\vartheta } \right]} . \\ \end{array} \end{equation}
Enmφϑ=mOnm=k=0nCnm,kdPnk(cosϑ)dϑϑφϑcoskφkPnk(cosϑ)sinkφφφϑ,Onmφϑ=mEnm=k=1nSnm,kdPnk(cosϑ)dϑϑφϑsinkφ+kPnk(cosϑ)coskφφφϑ.
(35)

From equations (22) one gets:

\begin{equation} \begin{array}{l} \left( {\displaystyle\frac{{\partial \vartheta '}}{{\partial \varphi }}} \right)_\vartheta = \sin \beta \sin \varphi ' , \\[10pt] \left( {\displaystyle\frac{{\partial \varphi '}}{{\partial \varphi }}} \right)_\vartheta = \cos \beta + \sin \beta \displaystyle\frac{{\cos \vartheta '}}{{\sin \vartheta '}}\cos \varphi ' . \\ \end{array} \end{equation}
ϑφϑ=sinβsinφ,φφϑ=cosβ+sinβcosϑsinϑcosφ.
(36)

Plugging (36) into (35), using formulas (A5), and identifying the term to term lead to the following remarkable crossed relations:

\begin{equation} \begin{array}{l} mS_n^{m,k} = k\cos \beta C_n^{m,k} + \displaystyle\frac{1}{2}\sin \beta \left[ {C_n^{m,k - 1} + \left( {n - k} \right)\left( {n + k + 1} \right)C_n^{m,k + 1} } \right] , \\[10pt] mC_n^{m,k} = k\cos \beta S_n^{m,k} + \displaystyle\frac{1}{2}\sin \beta \left[ {S_n^{m,k - 1} + \left( {n - k} \right)\left( {n + k + 1} \right)S_n^{m,k + 1} } \right] . \\ \end{array} \end{equation}
mSnm,k=kcosβCnm,k+12sinβCnm,k1+nkn+k+1Cnm,k+1,mCnm,k=kcosβSnm,k+12sinβSnm,k1+nkn+k+1Snm,k+1.
(37)

N.B.: In these relations, any |$C_\nu ^{\mu,\kappa } $|Cνμ,κ which does not respect 0 ⩽ μ ⩽ ν and 0 ⩽ κ ⩽ ν must be considered as zero as well as any |$S_\nu ^{\mu,\kappa } $|Sνμ,κ which does not respect 1 ⩽ μ ⩽ ν and 1 ⩽ κ ⩽ ν.

As a first example of the interest of such crossed relations, applying the second of (37) gives:

\begin{equation} mC_n^{m,0} = \displaystyle\frac{{n\left( {n + 1} \right)}}{2}\sin \beta S_n^{m,1} \end{equation}
mCnm,0=nn+12sinβSnm,1
(38)

Explicit formulas for the matrix coefficients result directly from the addition theorems (4) and (6) which are rewritten below in terms of relations (22) variables:

\begin{equation} P_n \left( {\cos \vartheta } \right) = P_n (\cos \beta)P_n (\cos \vartheta ') + 2\displaystyle\sum\limits_{k = 1}^n {\left( { - 1} \right)^k \displaystyle\frac{{\left( {n - k} \right)!}}{{\left( {n + k} \right)!}}P_n^k (\cos \beta)P_n^k (\cos \vartheta ')\cos m\varphi '}. \end{equation}
Pncosϑ=Pn(cosβ)Pn(cosϑ)+2k=1n1knk!n+k!Pnk(cosβ)Pnk(cosϑ)cosmφ.
(39)
\begin{equation} \begin{array}{l} P_n^m \left( {\cos \vartheta } \right) = \left( { - 1} \right)^m 2^m \left( {m - 1} \right)!\left( {\displaystyle\frac{{\sin \vartheta }}{{\sin \beta \sin \vartheta '}}} \right)^m \\[10pt] \times \displaystyle\sum\limits_{k = m}^n {\left( { - 1} \right)^k k\displaystyle\frac{{\left( {n - k} \right)!}}{{\left( {n + k} \right)!}}P_n^k (\cos \beta)P_n^k (\cos \vartheta ')C_{k - m}^m (\cos \varphi ')} . \\ \end{array} \end{equation}
Pnmcosϑ=1m2mm1!sinϑsinβsinϑm×k=mn1kknk!n+k!Pnk(cosβ)Pnk(cosϑ)Ckmm(cosφ).
(40)

Equation (39) gives immediately the first row of the Cn matrix and that of CSn through (25) and (14):

\begin{equation} \begin{array}{l} C_n^{0,0} = P_n (\cos \beta) , \\[10pt] C_n^{0,k} = \left( { - 1} \right)^k 2\displaystyle\frac{{\left( {n - k} \right)!}}{{\left( {n + k} \right)!}}P_n^k (\cos \beta) , \\[10pt] C_{Sn}^{0,k} = \left( { - 1} \right)^k P_{Sn}^k (\cos \beta) . \\ \end{array} \end{equation}
Cn0,0=Pn(cosβ),Cn0,k=1k2nk!n+k!Pnk(cosβ),CSn0,k=1kPSnk(cosβ).
(41)

Similarly, the first row of the Sn matrix is obtained by setting m = 1 in (40):

\begin{equation} \begin{array}{l} P_n^1 \left( {\cos \vartheta } \right) = - 2{\kern 1pt} \left( {\displaystyle\frac{{\sin \vartheta }}{{\sin \beta \sin \vartheta '}}} \right)\displaystyle\sum\limits_{k = 1}^n {\left( { - 1} \right)^k k\displaystyle\frac{{\left( {n - k} \right)!}}{{\left( {n + k} \right)!}}P_n^k (\cos \beta)P_n^k (\cos \vartheta ')C_{k - 1}^1 (\cos \varphi ')} , \\[10pt] C_{k - 1}^1 (\cos \varphi ') = \displaystyle\frac{{\sin k\varphi '}}{{\sin \varphi '}} , \\[10pt] P_n^1 \left( {\cos \vartheta } \right)\sin \varphi = - 2\displaystyle\sum\limits_{k = 1}^n {\left( { - 1} \right)^k k\displaystyle\frac{{\left( {n - k} \right)!}}{{\left( {n + k} \right)!}}P_n^k (\cos \beta)P_n^k (\cos \vartheta ')\sin k\varphi ' .} \\ \end{array} \end{equation}
Pn1cosϑ=2sinϑsinβsinϑk=1n1kknk!n+k!Pnk(cosβ)Pnk(cosϑ)Ck11(cosφ),Ck11(cosφ)=sinkφsinφ,Pn1cosϑsinφ=2k=1n1kknk!n+k!Pnk(cosβ)Pnk(cosϑ)sinkφ.
(42)

Hence

\begin{equation} S_n^{1,k} = - \left( { - 1} \right)^k 2k\displaystyle\frac{{\left( {n - k} \right)!}}{{\left( {n + k} \right)!}}\displaystyle\frac{{P_n^k (\cos \beta)}}{{\sin \beta }} , \end{equation}
Sn1,k=1k2knk!n+k!Pnk(cosβ)sinβ,
(43)

which could have been obtained immediately from (41) by using (33) and the crossed relation (38).

Explicit formulas for the subsequent rows of the Cn matrix are obtained by setting φ = φ′ = 0 and producing two expressions of |$P_n^m (\cos \vartheta)$|Pnm(cosϑ) through relations (23) and (40), those for the Sn matrix being further obtained through crossed relations (37). The proof develops as follows:

\begin{equation} \begin{array}{l} P_n^m (\cos \vartheta) = \displaystyle\sum\limits_{k = 0}^n {C_n^{m,k} P_n^k (\cos \vartheta ')} \\[10pt] = 2^m \left( {m - 1} \right)!\left( {\displaystyle\frac{{\sin \vartheta }}{{\sin \beta \sin \vartheta '}}} \right)^m \displaystyle\sum\limits_{k' = m}^n {\left( { - 1} \right)^{m + k'} k'\displaystyle\frac{{\left( {n - k'} \right)!}}{{\left( {n + k'} \right)!}}P_n^{k'} (\cos \beta)P_n^{k'} (\cos \vartheta ')C_{k' - m}^m (1)} , \\[10pt] \left( {\displaystyle\frac{{\sin \vartheta }}{{\sin \beta \sin \vartheta '}}} \right)^m = \left( {\displaystyle\frac{{\cos \beta }}{{\sin \beta }} + \displaystyle\frac{{\cos \vartheta '}}{{\sin \vartheta '}}} \right)^m = \displaystyle\sum\limits_{p = 0}^m {\left( {\begin{array}{*{20}c} m \\[10pt] p \\ \end{array}} \right)\left( {\displaystyle\frac{{\cos \beta }}{{\sin \beta }}} \right)^{m - p} \left( {\displaystyle\frac{{\cos \vartheta '}}{{\sin \vartheta '}}} \right)^p } , \\[10pt] C_{k' - m}^m (1) = \left( {\begin{array}{*{20}c} {k' + m - 1} \\[10pt] {k' - m} \\ \end{array}} \right) . \\ \end{array} \end{equation}
Pnm(cosϑ)=k=0nCnm,kPnk(cosϑ)=2mm1!sinϑsinβsinϑmk=mn1m+kknk!n+k!Pnk(cosβ)Pnk(cosϑ)Ckmm(1),sinϑsinβsinϑm=cosβsinβ+cosϑsinϑm=p=0mmpcosβsinβmpcosϑsinϑp,Ckmm(1)=k+m1km.
(44)

|$\left( {\displaystyle\frac{{\cos \beta }}{{\sin \beta }}} \right)^{m - p} P_n^{k^\prime } (\cos \beta)$|cosβsinβmpPnk(cosβ) and |$\left( {\displaystyle\frac{{\cos \vartheta ^\prime }}{{\sin \vartheta ^\prime }}} \right)^p P_n^{k^\prime } (\cos \vartheta ^\prime)$|cosϑsinϑpPnk(cosϑ) are then transformed by using appropriate relations of set (A4) in order to express the second form of |$P_n^m (\cos \vartheta)$|Pnm(cosϑ) in (44) with rational numbers and |$P_n^\mu (\cos \beta)P_n^{\mu ^\prime } (\cos \vartheta ^\prime)$|Pnμ(cosβ)Pnμ(cosϑ) products. Finally, |$C_n^{m,k} $|Cnm,k identifies with the coefficient of |$P_n^k (\cos \vartheta ^\prime)$|Pnk(cosϑ) in this second form.

Even with the help of Maple software, such a procedure is really cumbersome. Nevertheless, we succeeded in obtaining explicit formulas up to m = 4, that is the full Cn matrices for n ⩽ 4 and the five first rows of higher degree Cn, together with the first five columns by using symmetry properties (33). The lower triangle of these columns (0 ⩽ m ⩽ 4) is displayed below with that of the Sn matrices (1 ⩽ m ⩽ 3) obtained through the crossed relations (37):

\begin{equation} \begin{array}{l} C_n^{m \ge 1,1} = \displaystyle\frac{{\left( {n - 1} \right)!}}{{\left( {n + 1} \right)!}}\left[ {\displaystyle\frac{{\left( {n + m} \right)!\left( {n - m + 1} \right)!}}{{\left( {n - m} \right)!\left( {n + m - 1} \right)!}}P_n^{m - 1} (\cos \beta) - P_n^{\left( {m + 1} \right) \le n} (\cos \beta)} \right] , \\[15pt] S_n^{m \ge 1,1} = 2\displaystyle\frac{{\left( {n - 1} \right)!}}{{\left( {n + 1} \right)!}}\displaystyle\frac{{mP_n^m (\cos \beta)}}{{\sin \beta }} . \\ \end{array} \end{equation}
Cnm1,1=n1!n+1!n+m!nm+1!nm!n+m1!Pnm1(cosβ)Pnm+1n(cosβ),Snm1,1=2n1!n+1!mPnm(cosβ)sinβ.
(45)
\begin{equation} \begin{array}{l} C_n^{m \ge 2,2} = \displaystyle\frac{{\left( {n - 2} \right)!}}{{\left( {n + 2} \right)!}}\left[ \begin{array}{l} \displaystyle\frac{{\left( {n + m} \right)!\left( {n - m + 2} \right)!}}{{\left( {n - m} \right)!\left( {n + m - 2} \right)!}}P_n^{m - 2} (\cos \beta) \\[15pt] + 2m^2 P_n^m (\cos \beta) + P_n^{\left( {m + 2} \right) \le n} (\cos \beta) \\ \end{array} \right] , \\[24pt] S_n^{m \ge 2,2} = 2\displaystyle\frac{{\left( {n - 2} \right)!}}{{\left( {n + 2} \right)!}}\left[ \begin{array}{l} \displaystyle\frac{{\left( {n + m} \right)!\left( {n - m + 1} \right)!}}{{\left( {n - m} \right)!\left( {n + m - 1} \right)!}}\displaystyle\frac{{\left( {m - 1} \right)P_n^{m - 1} (\cos \beta)}}{{\sin \beta }} \\[15pt] - \displaystyle\frac{{\left( {m + 1} \right)P_n^{\left( {m + 1} \right) \le n} (\cos \beta)}}{{\sin \beta }} \end{array} \right] . \\ \end{array} \end{equation}
Cnm2,2=n2!n+2!n+m!nm+2!nm!n+m2!Pnm2(cosβ)+2m2Pnm(cosβ)+Pnm+2n(cosβ),Snm2,2=2n2!n+2!n+m!nm+1!nm!n+m1!m1Pnm1(cosβ)sinβm+1Pnm+1n(cosβ)sinβ.
(46)
\begin{equation} \begin{array}{l} C_n^{m \ge 3,3} = \displaystyle\frac{{\left( {n - 3} \right)!}}{{\left( {n + 3} \right)!}}\left[ \begin{array}{l} \displaystyle\frac{{\left( {n + m} \right)!\left( {n - m + 3} \right)!}}{{\left( {n - m} \right)!\left( {n + m - 3} \right)!}}P_n^{m - 3} (\cos \beta) \\[15pt] + 3m\left( {m - 1} \right)\displaystyle\frac{{\left( {n + m} \right)!\left( {n - m + 1} \right)!}}{{\left( {n - m} \right)!\left( {n + m - 1} \right)!}}P_n^{m - 1} (\cos \beta) \\[15pt] - 3m\left( {m + 1} \right)P_n^{\left( {m + 1} \right) \le n} (\cos \beta) - P_n^{\left( {m + 3} \right) \le n} (\cos \beta) \\ \end{array} \right] , \\[40pt] S_n^{m \ge 3,3} = 2\displaystyle\frac{{\left( {n - 3} \right)!}}{{\left( {n + 3} \right)!}}\left[ \begin{array}{l} \displaystyle\frac{{\left( {n + m} \right)!\left( {n - m + 2} \right)!}}{{\left( {n - m} \right)!\left( {n + m - 2} \right)!}}\displaystyle\frac{{\left( {m - 2} \right)P_n^{m - 2} (\cos \beta)}}{{\sin \beta }} \\[15pt] + \left( {2m^2 - 2 - n^2 - n} \right)\displaystyle\frac{{mP_n^{m - 1} (\cos \beta)}}{{\sin \beta }} \\[15pt] + \displaystyle\frac{{\left( {m + 2} \right)P_n^{\left( {m + 2} \right) \le n} (\cos \beta)}}{{\sin \beta }} \end{array} \right] . \\ \end{array} \end{equation}
Cnm3,3=n3!n+3!n+m!nm+3!nm!n+m3!Pnm3(cosβ)+3mm1n+m!nm+1!nm!n+m1!Pnm1(cosβ)3mm+1Pnm+1n(cosβ)Pnm+3n(cosβ),Snm3,3=2n3!n+3!n+m!nm+2!nm!n+m2!m2Pnm2(cosβ)sinβ+2m22n2nmPnm1(cosβ)sinβ+m+2Pnm+2n(cosβ)sinβ.
(47)
\begin{equation} C_n^{m \ge 4,4} = \displaystyle\frac{{\left( {n - 4} \right)!}}{{\left( {n + 4} \right)!}}\left[ \begin{array}{l} \displaystyle\frac{{\left( {n + m} \right)!\left( {n - m + 4} \right)!}}{{\left( {n - m} \right)!\left( {n + m - 4} \right)!}}P_n^{m - 4} (\cos \beta) \\[10pt] + 4m\left( {m - 2} \right)\displaystyle\frac{{\left( {n + m} \right)!\left( {n - m + 2} \right)!}}{{\left( {n - m} \right)!\left( {n + m - 2} \right)!}}P_n^{m - 2} (\cos \beta) \\[10pt] + 2m^2 \left( {3m^2 - 3 - 2n^2 - 2n} \right)P_n^m (\cos \beta) \\[10pt] + 4m\left( {m + 2} \right)P_n^{\left( {m + 2} \right) \le n} (\cos \beta) + P_n^{\left( {m + 4} \right) \le n} (\cos \beta) \\ \end{array} \right] . \end{equation}
Cnm4,4=n4!n+4!n+m!nm+4!nm!n+m4!Pnm4(cosβ)+4mm2n+m!nm+2!nm!n+m2!Pnm2(cosβ)+2m23m232n22nPnm(cosβ)+4mm+2Pnm+2n(cosβ)+Pnm+4n(cosβ).
(48)

It will be shown in the next section that the recurrence relations lead also to an explicit formula for the last row of the Cn and Sn matrices at any degree n.

The lower degree matrices are as follows (the argument cos β is omitted in |$P_\nu ^\mu $|Pνμ or |$P_{S\nu }^\mu $|PSνμ and the Legendre polynomials are noted |$P_\nu ^0 = P_{S\nu }^0 $|Pν0=PSν0 for the sake of homogeneity):

\begin{equation} C_1 = C_{S1} = \left[ {\begin{array}{*{20}c} {P_1^0 } & { - P_1^1 } \\[10pt] {P_1^1 } & {P_1^0 } \\ \end{array}} \right] ,\;S_1 = S_{S1} = \displaystyle\frac{1}{{\sin \beta }}\left[ {P_1^1 } \right] \end{equation}
C1=CS1=P10P11P11P10,S1=SS1=1sinβP11
(49)
\begin{equation} C_2 = \left[ {\begin{array}{c@\quad c@\quad c} {P_2^0 } & { - \displaystyle\frac{1}{3}P_2^1 } & {\displaystyle\frac{1}{{12}}P_2^2 } \\[10pt] {P_2^1 } & {P_2^0 - \displaystyle\frac{1}{6}P_2^2 } & { - \displaystyle\frac{1}{6}P_2^1 } \\[10pt] {P_2^2 } & {\displaystyle\frac{2}{3}P_2^1 } & {P_2^0 + \displaystyle\frac{1}{3}P_2^2 } \\ \end{array}} \right] ,\;S_2 = \displaystyle\frac{1}{{3\sin \beta }}\left[ {\begin{array}{c@\quad c} {P_2^1 } & { - \displaystyle\frac{1}{2}P_2^2 } \\[10pt] {2P_2^2 } & {P_2^1 } \\ \end{array}} \right] \end{equation}
C2=P2013P21112P22P21P2016P2216P21P2223P21P20+13P22,S2=13sinβP2112P222P22P21
(50)
\begin{equation} C_{S2} = \left[ {\begin{array}{c@\quad c@\quad c} {P_{S2}^0 } & { - P_{S2}^1 } & {P_{S2}^2 } \\[10pt] {P_{S2}^1 } & {P_{S2}^0 - \displaystyle\frac{1}{3}\sqrt 3 P_{S2}^2 } & { - \displaystyle\frac{1}{3}\sqrt 3 P_{S2}^1 } \\[10pt] {P_{S2}^2 } & {\displaystyle\frac{1}{3}\sqrt 3 P_{S2}^1 } & {P_{S2}^0 + \displaystyle\frac{2}{3}\sqrt 3 P_{S2}^2 } \\ \end{array}} \right] ,\;S_{S2} = \displaystyle\frac{1}{{\sqrt 3 \sin \beta }}\left[ {\begin{array}{c@\quad c} {P_{S2}^1 } & { - 2P_{S2}^2 } \\[10pt] {2P_{S2}^2 } & {P_{S2}^1 } \\ \end{array}} \right] \end{equation}
CS2=PS20PS21PS22PS21PS20133PS22133PS21PS22133PS21PS20+233PS22,SS2=13sinβPS212PS222PS22PS21
(51)

The general expressions for these matrices are

\begin{equation} \begin{array}{l} C_n^{m,k} (\beta) = \displaystyle\sum\limits_{p = \max \left( {0,k - m} \right)}^{\min \left( {k,\left[ {\frac{{n - m + k}}{2}} \right]} \right)} {R_C \left( {n,m,k,\mu = m - k + 2p} \right)P_n^\mu (\cos \beta)} \\[15pt] S_n^{m,k} (\beta) = \displaystyle\sum\limits_{q = \max \left( {0,k - m} \right)}^{\min \left( {k - 1,\left[ {\frac{{n - m + k - 1}}{2}} \right]} \right)} {R_S \left( {n,m,k,\mu ' = m - k + 2q + 1} \right)\displaystyle\frac{{P_n^{\mu '} (\cos \beta)}}{{\sin \beta }}} \end{array} \end{equation}
Cnm,k(β)=p=max0,kmmink,nm+k2RCn,m,k,μ=mk+2pPnμ(cosβ)Snm,k(β)=q=max0,kmmink1,nm+k12RSn,m,k,μ=mk+2q+1Pnμ(cosβ)sinβ
(52)

where RC and RS are rational numbers depending on the four integers in their parenthesis. The quasi-normalized matrices have equivalent expressions with |$R_{C_S } $|RCS and |$R_{S_S } $|RSS being square roots of rational numbers.

Trigonometric expressions of matrix coefficients, generated at any desired degree by a Maple program as explained in Section III B, can be transformed by another Maple program to be given the unique form (52). This program is a little more complicated than the previous one since it involves solving a set of polynomial equations for each element. The four-index R tables can thus be calculated once and for all and stored for further use. This has been done up to the ninth degree, in particular for checking examples of the explicit formulas and the recurrence relations.

Despite the academic interest inthe above explicit formulas and the capabilities of modern computer algebra software to build rotation matrix tables, the genuine and comprehensive solution to the problem of RSH rotation relies on recurrence relations. For clarity, detailed proofs are given separately for both kinds of RSH.

1. Even harmonics

The method consists of transforming the definition (23) of an even RSH in two different ways and identifying the final results. The first way simply makes use of relation (A1) to achieve:

\begin{equation} \begin{array}{l} \left( {2n + 1} \right)\cos \vartheta \displaystyle\sum\limits_{k = 0}^n {C_n^{m,k} P_n^k (\cos \vartheta ')\cos k\varphi '} \\[10pt] = \left( {n - m + 1} \right)\displaystyle\sum\limits_{k = 0}^{n + 1} {C_{n + 1}^{m,k} P_{n + 1}^k (\cos \vartheta ')\cos k\varphi ' + \left( {n + m} \right)\displaystyle\sum\limits_{k = 0}^{n - 1} {C_{n - 1}^{m,k} P_{n - 1}^k (\cos \vartheta ')\cos k\varphi '} } . \\ \end{array} \end{equation}
2n+1cosϑk=0nCnm,kPnk(cosϑ)coskφ=nm+1k=0n+1Cn+1m,kPn+1k(cosϑ)coskφ+n+mk=0n1Cn1m,kPn1k(cosϑ)coskφ.
(53)

One then makes cos ϑ enter the sum in the left part of (53) after replacing it by its expression from (22). One then makes use of relations (A1)–(A3) and shifts some indices to get:

\begin{equation} \begin{array}{l} \left( {2n + 1} \right)\displaystyle\sum\limits_{k = 0}^n {C_n^{m,k} \left( {\cos \beta \cos \vartheta ' - \sin \beta \sin \vartheta '\cos \varphi '} \right)P_n^k (\cos \vartheta ')\cos k\varphi '} \\[10pt] = \cos \beta \displaystyle\sum\limits_{k = 0}^n {\left( {n - k + 1} \right)C_n^{m,k} P_{n + 1}^k (\cos \vartheta ')\cos k\varphi '} \\[10pt] + \cos \beta \displaystyle\sum\limits_{k = 0}^{n - 1} {\left( {n + k} \right)C_n^{m,k} P_{n - 1}^k (\cos \vartheta ')\cos k\varphi '} \\[10pt] - \sin \beta C_n^{m,0} \left[ {P_{n + 1}^1 (\cos \vartheta ') - P_{n - 1}^1 (\cos \vartheta ')} \right]\cos \varphi ' \\[10pt] - \displaystyle\frac{1}{2}\sin \beta \displaystyle\sum\limits_{k = 2}^{n + 1} {C_n^{m,k - 1} \left[ {P_{n + 1}^k (\cos \vartheta ') - P_{n - 1}^k (\cos \vartheta ')} \right]} \cos k\varphi ' \\[10pt] - \displaystyle\frac{1}{2}\sin \beta \displaystyle\sum\limits_{k = 0}^{n - 1} {C_n^{m,k + 1} \left[ \begin{array}{c} \left( {n + k} \right)\left( {n + k + 1} \right)P_{n - 1}^k (\cos \vartheta ') \\[10pt] - \left( {n - k} \right)\left( {n - k + 1} \right)P_{n + 1}^k (\cos \vartheta ') \\ \end{array} \right]} \cos k\varphi ' \\ \end{array} \end{equation}
2n+1k=0nCnm,kcosβcosϑsinβsinϑcosφPnk(cosϑ)coskφ=cosβk=0nnk+1Cnm,kPn+1k(cosϑ)coskφ+cosβk=0n1n+kCnm,kPn1k(cosϑ)coskφsinβCnm,0Pn+11(cosϑ)Pn11(cosϑ)cosφ12sinβk=2n+1Cnm,k1Pn+1k(cosϑ)Pn1k(cosϑ)coskφ12sinβk=0n1Cnm,k+1n+kn+k+1Pn1k(cosϑ)nknk+1Pn+1k(cosϑ)coskφ
(54)

N.B.: In all these proofs, any |$P_\nu ^\mu $|Pνμ which does not respect 0 ⩽ μ ⩽ ν must be considered as zero.

Since RSH are orthogonal functions, one gets from (53) and (54) two recurrence relations by identifying the coefficients of |$E_{n + 1}^k ( {\vartheta ^\prime,\varphi ^\prime } )$|En+1k(ϑ,φ) on the one hand and |$E_{n - 1}^k ( {\vartheta ^\prime,\varphi ^\prime } )$|En1k(ϑ,φ) on the other:

\begin{equation} \hspace*{-8pt}\begin{array}{l} \left( {n - m} \right)C_n^{m,k} = - \displaystyle\frac{{1 + \delta _k^1 }}{2}\sin \beta C_{n - 1}^{m,k - 1} \\[10pt] + \cos \beta \left( {n - k} \right)C_{n - 1}^{m,k} + \displaystyle\frac{1}{2}\sin \beta \left( {n - k - 1} \right)\left( {n - k} \right)C_{n - 1}^{m,k + 1} \quad \forall 0 \le m \le n - 1,\;0 \le k \le n. \!\!\! \\ \end{array} \end{equation}
nmCnm,k=1+δk12sinβCn1m,k1+cosβnkCn1m,k+12sinβnk1nkCn1m,k+10mn1,0kn.
(55)
\begin{equation} \hspace*{-7pt}\begin{array}{l} \left( {n + m} \right)C_{n - 1}^{m,k} = \displaystyle\frac{{1 + \delta _k^1 }}{2}\sin \beta C_n^{m,k - 1} \\[10pt] + \cos \beta \left( {n + k} \right)C_n^{m,k} - \displaystyle\frac{1}{2}\sin \beta \left( {n + k} \right)\left( {n + k + 1} \right)C_n^{m,k + 1} \quad \forall 0 \le m \le n,\;0 \le k \le n - 1. \!\! \\ \end{array} \end{equation}
n+mCn1m,k=1+δk12sinβCnm,k1+cosβn+kCnm,k12sinβn+kn+k+1Cnm,k+10mn,0kn1.
(56)

N.B.: In all these relations, any |$C_\nu ^{\mu,\kappa } $|Cνμ,κ which does not respect 0 ⩽ μ ⩽ ν and 0 ⩽ κ ⩽ ν must be considered as zero. The subtle |$\delta _k^1 $|δk1 arises from shifting some k indices.

Other recurrences can be obtained by combining (55) and (56), for instance by shifting n to n + 1 in the first and adding side by side the second to eliminate |$\delta _k^1 $|δk1 resulting in:

\begin{equation} \left( {n - m + 1} \right)C_{n + 1}^{m,k} + \left( {n + m} \right)C_{n - 1}^{m,k} = \left( {2n + 1} \right)\left[ {\cos \beta C_n^{m,k} - k\sin \beta C_n^{m,k + 1} } \right] . \end{equation}
nm+1Cn+1m,k+n+mCn1m,k=2n+1cosβCnm,kksinβCnm,k+1.
(57)

By setting k = 0 in (57) one gets:

\begin{equation} \left( {n - m + 1} \right)C_{n + 1}^{m,0} + \left( {n + m} \right)C_{n - 1}^{m,0} = \left( {2n + 1} \right)\cos \beta C_n^{m,0} , \end{equation}
nm+1Cn+1m,0+n+mCn1m,0=2n+1cosβCnm,0,
(58)

which is none other than an associated Legendre function recurrence. Plugging in the known initial values it gives

\begin{equation} C_n^{m,0} = P_n^m (\cos \beta) . \end{equation}
Cnm,0=Pnm(cosβ).
(59)

Recurrence (55) allows any Cn matrix to be built from that of the lower degree Cn − 1 except for its last row m = n. But in that case, recurrence (56) gives the following answer:

\begin{equation} \begin{array}{l} \displaystyle\frac{{1 + \delta _k^1 }}{2}\sin \beta {\kern 1pt} {\kern 1pt} C_n^{n,k - 1} + \cos \beta \left( {n + k} \right)C_n^{n,k} - \displaystyle\frac{1}{2}\sin \beta {\kern 1pt} \left( {n + k} \right)\left( {n + k + 1} \right)C_n^{n,k + 1} = 0 , \\[10pt] C_n^{n,0} = P_n^n (\cos \beta) = \left( {2n - 1} \right)!!{\kern 1pt} {\kern 1pt} {\kern 1pt} \sin ^n \beta , \\[10pt] C_n^{n,1} = \displaystyle\frac{2}{{n + 1}}\displaystyle\frac{{\cos \beta }}{{\sin \beta }}C_n^{n,0} = \displaystyle\frac{2}{{n + 1}}\left( {2n - 1} \right)!!{\kern 1pt} {\kern 1pt} {\kern 1pt} \sin ^{n - 1} \beta \cos \beta = \displaystyle\frac{2}{{n + 1}}P_n^{n - 1} (\cos \beta) , \\[10pt] C_n^{n,2} = \displaystyle\frac{2}{{\left( {n + 1} \right)\left( {n + 2} \right)}}\left( {2n - 1} \right)!!\sin ^{n - 2} \beta \left( {1 + \cos ^2 \beta } \right) \\[10pt] = \displaystyle\frac{2}{{\left( {n - 1} \right)\left( {n + 1} \right)\left( {n + 2} \right)}}\left[ {2\left( {2n - 1} \right)P_n^{n - 2} (\cos \beta) + nP_n^n (\cos \beta)} \right] , \\[10pt] C_n^{n,k \ge 3} = \displaystyle\frac{1}{{n + k}}\left( {2\displaystyle\frac{{\cos \beta }}{{\sin \beta }}C_n^{n,k - 1} + \displaystyle\frac{1}{{n + k - 1}}C_n^{n,k - 2} } \right) . \\ \end{array} \end{equation}
1+δk12sinβCnn,k1+cosβn+kCnn,k12sinβn+kn+k+1Cnn,k+1=0,Cnn,0=Pnn(cosβ)=2n1!!sinnβ,Cnn,1=2n+1cosβsinβCnn,0=2n+12n1!!sinn1βcosβ=2n+1Pnn1(cosβ),Cnn,2=2n+1n+22n1!!sinn2β1+cos2β=2n1n+1n+222n1Pnn2(cosβ)+nPnn(cosβ),Cnn,k3=1n+k2cosβsinβCnn,k1+1n+k1Cnn,k2.
(60)

This recurrence can be solved by classical methods and leads to the following compact form:

\begin{equation} C_n^{n,k \ge 1} = \displaystyle\frac{1}{{2^{n - 1} }}\displaystyle\frac{{\left( {2n} \right)!}}{{\left( {n + k} \right)!}}\sin ^{n - k} \beta \displaystyle\sum\limits_{p = 0}^{\left[ {\frac{k}{2}} \right]} {\left( {\begin{array}{*{20}c} k \\[10pt] {2p} \\ \end{array}} \right)\cos ^{k - 2p} \beta } . \end{equation}
Cnn,k1=12n12n!n+k!sinnkβp=0k2k2pcosk2pβ.
(61)

By using the first equation of (A4) this formula is transformed to the following:

\begin{equation} \begin{array}{l} C_n^{n,k} = \displaystyle\frac{{\left( {n - k} \right)!}}{{\left( {n + k} \right)!}}\displaystyle\frac{{\left( {2n} \right)!k!}}{{\left( {2n - k} \right)!}}P_n^{n - k} (\cos \beta)\;\;0 \le k \le \min \left( {n,1} \right) , \\[10pt] C_n^{n,k \ge 2} = \displaystyle\frac{{\left( {n - k} \right)!}}{{\left( {n + k} \right)!}}\displaystyle\frac{{\left( {2n} \right)!k!}}{{\left( {2n - k} \right)!}}P_n^{n - k} (\cos \beta) \\[10pt] + k\displaystyle\frac{{\left( {2n} \right)!}}{{\left( {n - 1} \right)!}}\displaystyle\sum\limits_{p = 1}^{\left[ {\frac{k}{2}} \right]} {\displaystyle\frac{{\left( {k - p - 1} \right)!\left( {n + p - 1} \right)!\left( {n - k + p - 1} \right)!}}{{\left( {2n - k + 2p} \right)!\left( {n + k} \right)!p!}}\left( {n - k + 2p} \right)} P_n^{n - k + 2p} (\cos \beta) . \\ \end{array} \end{equation}
Cnn,k=nk!n+k!2n!k!2nk!Pnnk(cosβ)0kminn,1,Cnn,k2=nk!n+k!2n!k!2nk!Pnnk(cosβ)+k2n!n1!p=1k2kp1!n+p1!nk+p1!2nk+2p!n+k!p!nk+2pPnnk+2p(cosβ).
(62)

By setting k = n in (61) and (62) one gets an interesting (and possibly new) summation formula for associated Legendre functions:

\begin{equation} \begin{array}{l} P_n (\cos \beta) + \displaystyle\frac{{2n}}{{\left( {n - 1} \right)!}}\displaystyle\sum\limits_{p = 1}^{\left[ {\frac{n}{2}} \right]} {\displaystyle\frac{{\left( {n - p - 1} \right)!\left( {n + p - 1} \right)!}}{{\left( {n + 2p} \right)!}}P_n^{2p} (\cos \beta)} \\[10pt] = \displaystyle\frac{1}{{2^{n - 1} }}\displaystyle\sum\limits_{p = 0}^{\left[ {\frac{n}{2}} \right]} {\left( {\begin{array}{*{20}c} n \\[10pt] {2p} \\ \end{array}} \right)\cos ^{n - 2p} \beta } . \\ \end{array} \end{equation}
Pn(cosβ)+2nn1!p=1n2np1!n+p1!n+2p!Pn2p(cosβ)=12n1p=0n2n2pcosn2pβ.
(63)

Another interesting relation is obtained by transforming the right side of (55) with (56), which leads to a recurrence involving the single index k ⩾ 0:

\begin{equation} \begin{array}{l} \left( {1 + \delta _k^0 } \right)\sin ^2 \beta C_n^{m,k} + 2\left( {2k + 3} \right)\sin \beta \cos \beta C_n^{m,k + 1} \\[10pt] + 2\left\{ {\left[ {n\left( {n + 1} \right) - 3\left( {k + 2} \right)^2 } \right]\sin ^2 \beta - 2\left( {m - k - 2} \right)\left( {m + k + 2} \right)} \right\}C_n^{m,k + 2} \\[10pt] + 2\left( {2k + 5} \right)\left( {n - k - 2} \right)\left( {n + k + 3} \right)\sin \beta \cos \beta C_n^{m,k + 3} \\[10pt] + \left( {n - k - 3} \right)\left( {n - k - 2} \right)\left( {n + k + 3} \right)\left( {n + k + 4} \right)\sin ^2 \beta C_n^{m,k + 4} = 0 \\ \end{array} \end{equation}
1+δk0sin2βCnm,k+22k+3sinβcosβCnm,k+1+2nn+13k+22sin2β2mk2m+k+2Cnm,k+2+22k+5nk2n+k+3sinβcosβCnm,k+3+nk3nk2n+k+3n+k+4sin2βCnm,k+4=0
(64)

This equation is valid for n ⩾ 2 and must contain at least three terms |$C_n^{m,\kappa } $|Cnm,κ with 0 ⩽ κ ⩽ n, while terms with κ out of range are set to zero. For an arbitrary large degree, the three first left columns of the Cn matrix must be known in order to generate the whole matrix but they are precisely given by the explicit formulas of Section III D.

2. Odd harmonics

The proof follows exactly the same steps which are not reproduced here and the final formulas below are simpler than that for even harmonics since the complication of |$\delta _k^1 $|δk1 does not exist:

\begin{equation} \begin{array}{l} \left( {n - m} \right)S_n^{m,k} = - \displaystyle\frac{1}{2}\sin \beta S_{n - 1}^{m,k - 1} + \cos \beta \left( {n - k} \right)S_{n - 1}^{m,k} \\[10pt] + \displaystyle\frac{1}{2}\sin \beta \left( {n - k - 1} \right)\left( {n - k} \right)S_{n - 1}^{m,k + 1} \;\;\forall 1 \le m \le n - 1,\;1 \le k \le n . \\ \end{array} \end{equation}
nmSnm,k=12sinβSn1m,k1+cosβnkSn1m,k+12sinβnk1nkSn1m,k+11mn1,1kn.
(65)
\begin{equation} \begin{array}{l} \left( {n + m} \right)S_{n - 1}^{m,k} = \displaystyle\frac{1}{2}\sin \beta S_n^{m,k - 1} + \cos \beta \left( {n + k} \right)S_n^{m,k} \\[10pt] - \displaystyle\frac{1}{2}\sin \beta \left( {n + k} \right)\left( {n + k + 1} \right)S_n^{m,k + 1} \;\;\forall 1 \le m \le n,\;1 \le k \le n - 1 . \\ \end{array} \end{equation}
n+mSn1m,k=12sinβSnm,k1+cosβn+kSnm,k12sinβn+kn+k+1Snm,k+11mn,1kn1.
(66)

N.B.: In all these relations, any |$S_\nu ^{\mu,\kappa } $|Sνμ,κ which does not respect 1 ⩽ μ ⩽ ν and 1 ⩽ κ ⩽ ν must be considered as zero.

\begin{equation} \begin{array}{l} S_n^{n,1} = \displaystyle\frac{2}{{n + 1}}\displaystyle\frac{{P_n^n (\cos \beta)}}{{\sin \beta }} = \displaystyle\frac{2}{{n + 1}}\left( {2n - 1} \right)!!\sin ^{n - 1} \beta , \\[10pt] S_n^{n,2} = \displaystyle\frac{4}{{\left( {n + 1} \right)\left( {n + 2} \right)}}\displaystyle\frac{{P_n^{n - 1} (\cos \beta)}}{{\sin \beta }} = \displaystyle\frac{4}{{\left( {n + 1} \right)\left( {n + 2} \right)}}\left( {2n - 1} \right)!!\cos \beta \sin ^{n - 2} \beta , \\[10pt] S_n^{n,k \ge 2} = \displaystyle\frac{1}{{n + k}}\left( {2\displaystyle\frac{{\cos \beta }}{{\sin \beta }}S_n^{n,k - 1} + \displaystyle\frac{1}{{n + k - 1}}S_n^{n,k - 2} } \right) . \\ \end{array} \end{equation}
Snn,1=2n+1Pnn(cosβ)sinβ=2n+12n1!!sinn1β,Snn,2=4n+1n+2Pnn1(cosβ)sinβ=4n+1n+22n1!!cosβsinn2β,Snn,k2=1n+k2cosβsinβSnn,k1+1n+k1Snn,k2.
(67)
\begin{equation} S_n^{n,k \ge 1} = \displaystyle\frac{1}{{2^{n - 1} }}\displaystyle\frac{{\left( {2n} \right)!}}{{\left( {n + k} \right)!}}\sin ^{n - k} \beta \displaystyle\sum\limits_{p = 0}^{\left[ {\frac{{k - 1}}{2}} \right]} {\left( {\begin{array}{*{20}c} k \\[10pt] {2p + 1} \\ \end{array}} \right)\cos ^{k - 2p - 1} \beta } . \end{equation}
Snn,k1=12n12n!n+k!sinnkβp=0k12k2p+1cosk2p1β.
(68)
\begin{equation} \begin{array}{l} S_n^{n,k} = 2\displaystyle\frac{{\left( {2n} \right)!}}{{n!}} \\[10pt] \times \displaystyle\sum\limits_{p = 0}^{\left[ {\frac{{k - 1}}{2}} \right]} {\displaystyle\frac{{\left( {k - p - 1} \right)!\left( {n + p} \right)!\left( {n - k + p} \right)!}}{{\left( {2n - k + 2p + 1} \right)!\left( {n + k} \right)!p!}}\left( {n - k + 2p + 1} \right)} \displaystyle\frac{{P_n^{n - k + 2p + 1} (\cos \beta)}}{{\sin \beta }} . \\ \end{array} \end{equation}
Snn,k=22n!n!×p=0k12kp1!n+p!nk+p!2nk+2p+1!n+k!p!nk+2p+1Pnnk+2p+1(cosβ)sinβ.
(69)

And finally, another summation formula for associated Legendre functions:

\begin{equation} \displaystyle\frac{2}{{n!}}\displaystyle\sum\limits_{p = 0}^{\left[ {\frac{{n - 1}}{2}} \right]} {\displaystyle\frac{{\left( {n - p - 1} \right)!\left( {n + p} \right)!}}{{\left( {n + 2p + 1} \right)!}}\left( {2p + 1} \right)\displaystyle\frac{{P_n^{2p + 1} (\cos \beta)}}{{\sin \beta }}} = \displaystyle\frac{1}{{2^{n - 1} }}\displaystyle\sum\limits_{p = 0}^{\left[ {\frac{{n - 1}}{2}} \right]} {\left( {\begin{array}{*{20}c} n \\[10pt] {2p + 1} \\ \end{array}} \right)\cos ^{n - 2p - 1} \beta } . \end{equation}
2n!p=0n12np1!n+p!n+2p+1!2p+1Pn2p+1(cosβ)sinβ=12n1p=0n12n2p+1cosn2p1β.
(70)

The recurrence similar to (64) involving the single index k ⩾ 1 is as follows:

\begin{equation} \begin{array}{l} \sin ^2 \beta S_n^{m,k} + 2\left( {2k + 3} \right)\sin \beta \cos \beta S_n^{m,k + 1} \\[15pt] + 2\left\{ {\left[ {n\left( {n + 1} \right) - 3\left( {k + 2} \right)^2 } \right]\sin ^2 \beta - 2\left( {m - k - 2} \right)\left( {m + k + 2} \right)} \right\}S_n^{m,k + 2} \\[15pt] + 2\left( {2k + 5} \right)\left( {n - k - 2} \right)\left( {n + k + 3} \right)\sin \beta \cos \beta S_n^{m,k + 3} \\[15pt] + \left( {n - k - 3} \right)\left( {n - k - 2} \right)\left( {n + k + 3} \right)\left( {n + k + 4} \right)\sin ^2 \beta S_n^{m,k + 4} = 0 \\ \end{array} \end{equation}
sin2βSnm,k+22k+3sinβcosβSnm,k+1+2nn+13k+22sin2β2mk2m+k+2Snm,k+2+22k+5nk2n+k+3sinβcosβSnm,k+3+nk3nk2n+k+3n+k+4sin2βSnm,k+4=0
(71)

This equation is valid for n ⩾ 2 and must contain at least three terms |$S_n^{m,\kappa } $|Snm,κ with 1 ⩽ κ ⩽ n, while terms with κ out of range are set to zero. For an arbitrary large degree, the three first left columns of the Sn matrix must be known in order to generate the whole matrix but they are precisely given by the explicit formulas of Section III D.

3. Relations for quasi-normalized Legendre functions and matrices

As explained in Section II C 2, it is much better for numerical stability to weight or normalize associated Legendre functions in a convenient way but the above recurrence relations must be rewritten accordingly. This rewriting is, in principle, straightforward but it can be a source of mistakes, in particular with square roots if Schmidt quasi-normalization is chosen. For this reason, the corresponding recurrence relations are explicitly given below.

\begin{equation} \begin{array}{l} \sqrt {\left( {n - m} \right)\left( {n + m} \right)} C_{Sn}^{m,0} = n\cos \beta C_{Sn - 1}^{m,0} + \sin \beta \sqrt {\displaystyle\frac{{n\left( {n - 1} \right)}}{2}} C_{Sn - 1}^{m,k + 1} , \\[15pt] \sqrt {\left( {n - m} \right)\left( {n + m} \right)} C_{Sn}^{m,1} = - \sin \beta \sqrt {\displaystyle\frac{{n\left( {n + 1} \right)}}{2}} C_{Sn - 1}^{m,0} \\[15pt] + \cos \beta \sqrt {\left( {n - 1} \right)\left( {n + 1} \right)} C_{Sn - 1}^{m,1} + \displaystyle\frac{1}{2}\sin \beta \sqrt {\left( {n - k - 1} \right)\left( {n - k} \right)} C_{Sn - 1}^{m,2} , \\[15pt] \sqrt {\left( {n - m} \right)\left( {n + m} \right)} C_{Sn}^{m,k} = - \displaystyle\frac{1}{2}\sin \beta \sqrt {\left( {n + k - 1} \right)\left( {n + k} \right)} C_{Sn - 1}^{m,k - 1} \\[15pt] + \cos \beta \sqrt {\left( {n - k} \right)\left( {n + k} \right)} C_{Sn - 1}^{m,k} + \displaystyle\frac{1}{2}\sin \beta \sqrt {\left( {n - k - 1} \right)\left( {n - k} \right)} C_{Sn - 1}^{m,k + 1} , \\[15pt] \forall 0 \le m \le n - 1,\;2 \le k \le n . \\ \end{array} \end{equation}
nmn+mCSnm,0=ncosβCSn1m,0+sinβnn12CSn1m,k+1,nmn+mCSnm,1=sinβnn+12CSn1m,0+cosβn1n+1CSn1m,1+12sinβnk1nkCSn1m,2,nmn+mCSnm,k=12sinβn+k1n+kCSn1m,k1+cosβnkn+kCSn1m,k+12sinβnk1nkCSn1m,k+1,0mn1,2kn.
(72)
\begin{equation} \begin{array}{l} \sqrt {\left( {n - m} \right)\left( {n + m} \right)} S_{Sn}^{m,k} = - \displaystyle\frac{1}{2}\sin \beta \sqrt {\left( {n + k - 1} \right)\left( {n + k} \right)} {\kern 1pt} S_{Sn - 1}^{m,k - 1} \\[15pt] + \cos \beta \sqrt {\left( {n - k} \right)\left( {n + k} \right)} S_{Sn - 1}^{m,k} + \displaystyle\frac{1}{2}\sin \beta \sqrt {\left( {n - k - 1} \right)\left( {n - k} \right)} S_{Sn - 1}^{m,k + 1} , \\[15pt] \forall 1 \le m \le n - 1,\;1 \le k \le n . \\ \end{array} \end{equation}
nmn+mSSnm,k=12sinβn+k1n+kSSn1m,k1+cosβnkn+kSSn1m,k+12sinβnk1nkSSn1m,k+1,1mn1,1kn.
(73)

The CSn and SSn matrices’ first columns are:

\begin{equation} \begin{array}{l} C_{Sn}^{m,0} = P_{Sn}^m (\cos \beta) , \\[15pt] S_{Sn}^{m,1} = m\sqrt {\displaystyle\frac{2}{{n\left( {n + 1} \right)}}} \displaystyle\frac{{P_{Sn}^m (\cos \beta)}}{{\sin \beta }} . \\ \end{array} \end{equation}
CSnm,0=PSnm(cosβ),SSnm,1=m2nn+1PSnm(cosβ)sinβ.
(74)

Their last rows are:

\begin{equation} \begin{array}{l} C_{Sn}^{n,k \ge 1} = \displaystyle\frac{1}{{2^{n - 1} }}\sqrt {\displaystyle\frac{{\left( {2n} \right)!}}{{\left( {n - k} \right)!\left( {n + k} \right)!}}} \sin ^{n - k} \beta {\kern 1pt} {\kern 1pt} \displaystyle\sum\limits_{p = 0}^{\left[ {\frac{k}{2}} \right]} {\left( {\begin{array}{*{20}c} n \\[15pt] {2p} \\ \end{array}} \right)\cos ^{k - 2p} \beta } , \\[15pt] C_{Sn}^{n,k} = \sqrt {\displaystyle\frac{2}{{2 - \delta _k^0 }}\displaystyle\frac{1}{{2 - \delta _{n - k}^0 }}\displaystyle\frac{{\left( {2n} \right)!\left( {n - k} \right)!k!}}{{\left( {n + k} \right)!\left( {2n - k} \right)!}}} P_{Sn}^{n - k} (\cos \beta) ,\;0 \le k \le \min \left( {n,1} \right) , \\[15pt] C_{Sn}^{n,k \ge 2} = \sqrt {\displaystyle\frac{1}{{2 - \delta _{n - k}^0 }}\displaystyle\frac{{\left( {2n} \right)!\left( {n - k} \right)!k!}}{{\left( {n + k} \right)!\left( {2n - k} \right)!}}} P_{Sn}^{n - k} (\cos \beta) + \displaystyle\frac{k}{{\left( {n - 1} \right)!}}\sqrt {\displaystyle\frac{{\left( {2n} \right)!}}{{\left( {n - k} \right)!\left( {n + k} \right)!}}} \\[15pt] \times \displaystyle\sum\limits_{p = 1}^{\left[ {\frac{k}{2}} \right]} {\left[ \begin{array}{c} \displaystyle\frac{{\left( {k - p - 1} \right)!\left( {n + p - 1} \right)!\left( {n - k + p - 1} \right)!}}{{p!}}\left( {n - k + 2p} \right) \times \\[15pt] \sqrt {\displaystyle\frac{1}{{2 - \delta _{n - k + 2p}^0 }}\displaystyle\frac{1}{{\left( {k - 2p} \right)!\left( {2n - k + 2p} \right)!}}} P_{Sn}^{n - k + 2p} (\cos \beta) \\ \end{array} \right]} . \\ \end{array} \end{equation}
CSnn,k1=12n12n!nk!n+k!sinnkβp=0k2n2pcosk2pβ,CSnn,k=22δk012δnk02n!nk!k!n+k!2nk!PSnnk(cosβ),0kminn,1,CSnn,k2=12δnk02n!nk!k!n+k!2nk!PSnnk(cosβ)+kn1!2n!nk!n+k!×p=1k2kp1!n+p1!nk+p1!p!nk+2p×12δnk+2p01k2p!2nk+2p!PSnnk+2p(cosβ).
(75)
\begin{equation} \begin{array}{l} S_{Sn}^{n,k \ge 1} = \displaystyle\frac{1}{{2^{n - 1} }}\sqrt {\displaystyle\frac{{\left( {2n} \right)!}}{{\left( {n - k} \right)!\left( {n + k} \right)!}}} \sin ^{n - k} \beta \displaystyle\sum\limits_{p = 0}^{\left[ {\frac{{k - 1}}{2}} \right]} {\left( {\begin{array}{*{20}c} n \\[15pt] {2p + 1} \\ \end{array}} \right)\cos ^{k - 2p - 1} \beta } , \\[15pt] S_{Sn}^{n,k} = \displaystyle\frac{{\sqrt 2 }}{{n!}}\sqrt {\displaystyle\frac{{\left( {2n} \right)!}}{{\left( {n - k} \right)!\left( {n + k} \right)!}}} \\[15pt] \times \displaystyle\sum\limits_{p = 0}^{\left[ {\frac{{k - 1}}{2}} \right]} {\left[ \begin{array}{c} \displaystyle\frac{{\left( {k - p - 1} \right)!\left( {n + p} \right)!\left( {n - k + p} \right)!}}{{p!}}\left( {n - k + 2p + 1} \right) \times \\[15pt] \sqrt {\displaystyle\frac{1}{{\left( {k - 2p - 1} \right)!\left( {2n - k + 2p + 1} \right)!}}} \displaystyle\frac{{P_n^{n - k + 2p + 1} (\cos \beta)}}{{\sin \beta }} \\ \end{array} \right]} . \\ \end{array} \end{equation}
SSnn,k1=12n12n!nk!n+k!sinnkβp=0k12n2p+1cosk2p1β,SSnn,k=2n!2n!nk!n+k!×p=0k12kp1!n+p!nk+p!p!nk+2p+1×1k2p1!2nk+2p+1!Pnnk+2p+1(cosβ)sinβ.
(76)

The four-element single-index recurrences are:

\begin{equation} \begin{array}{l} \sqrt {1 + \delta _k^0 } \sin ^2 \beta {\kern 1pt} C_{Sn}^{m,k} + 2\displaystyle\frac{{2k + 3}}{{\sqrt {\left( {n - k} \right)\left( {n + k + 1} \right)} }}\sin \beta \cos \beta {\kern 1pt} C_{Sn}^{m,k + 1} \\[15pt] + 2\displaystyle\frac{{\left[ {n\left( {n + 1} \right) - 3\left( {k + 2} \right)^2 } \right]\sin ^2 \beta - 2\left( {m - k - 2} \right)\left( {m + k + 2} \right)}}{{\sqrt {\left( {n - k - 1} \right)\left( {n - k} \right)\left( {n + k + 1} \right)\left( {n + k + 2} \right)} }}C_{Sn}^{m,k + 2} \\[15pt] + 2\left( {2k + 5} \right)\sqrt {\displaystyle\frac{{\left( {n - k - 2} \right)\left( {n + k + 3} \right)}}{{\left( {n - k - 1} \right)\left( {n - k} \right)\left( {n + k + 1} \right)\left( {n + k + 2} \right)}}} \sin \beta \cos \beta C_{Sn}^{m,k + 3} \\[15pt] + \sqrt {\displaystyle\frac{{\left( {n - k - 3} \right)\left( {n - k - 2} \right)\left( {n + k + 3} \right)\left( {n + k + 4} \right)}}{{\left( {n - k - 1} \right)\left( {n - k} \right)\left( {n + k + 1} \right)\left( {n + k + 2} \right)}}} \sin ^2 \beta C_{Sn}^{m,k + 4} = 0 \\ \end{array} \end{equation}
1+δk0sin2βCSnm,k+22k+3nkn+k+1sinβcosβCSnm,k+1+2nn+13k+22sin2β2mk2m+k+2nk1nkn+k+1n+k+2CSnm,k+2+22k+5nk2n+k+3nk1nkn+k+1n+k+2sinβcosβCSnm,k+3+nk3nk2n+k+3n+k+4nk1nkn+k+1n+k+2sin2βCSnm,k+4=0
(77)
\begin{equation} \begin{array}{l} \sin ^2 \beta {\kern 1pt} S_{Sn}^{m,k} + 2\displaystyle\frac{{2k + 3}}{{\sqrt {\left( {n - k} \right)\left( {n + k + 1} \right)} }}\sin \beta \cos \beta {\kern 1pt} S_{Sn}^{m,k + 1} \\[15pt] + 2\displaystyle\frac{{\left[ {n\left( {n + 1} \right) - 3\left( {k + 2} \right)^2 } \right]\sin ^2 \beta - 2\left( {m - k - 2} \right)\left( {m + k + 2} \right)}}{{\sqrt {\left( {n - k - 1} \right)\left( {n - k} \right)\left( {n + k + 1} \right)\left( {n + k + 2} \right)} }}S_{Sn}^{m,k + 2} \\[15pt] + 2\left( {2k + 5} \right)\sqrt {\displaystyle\frac{{\left( {n - k - 2} \right)\left( {n + k + 3} \right)}}{{\left( {n - k - 1} \right)\left( {n - k} \right)\left( {n + k + 1} \right)\left( {n + k + 2} \right)}}} \sin \beta \cos \beta S_{Sn}^{m,k + 3} \\[15pt] + \sqrt {\displaystyle\frac{{\left( {n - k - 3} \right)\left( {n - k - 2} \right)\left( {n + k + 3} \right)\left( {n + k + 4} \right)}}{{\left( {n - k - 1} \right)\left( {n - k} \right)\left( {n + k + 1} \right)\left( {n + k + 2} \right)}}} \sin ^2 \beta S_{Sn}^{m,k + 4} = 0 \\ \end{array} \end{equation}
sin2βSSnm,k+22k+3nkn+k+1sinβcosβSSnm,k+1+2nn+13k+22sin2β2mk2m+k+2nk1nkn+k+1n+k+2SSnm,k+2+22k+5nk2n+k+3nk1nkn+k+1n+k+2sinβcosβSSnm,k+3+nk3nk2n+k+3n+k+4nk1nkn+k+1n+k+2sin2βSSnm,k+4=0
(78)

4. Relations for the matrix coefficients in terms of Legendre functions

Since Cn and Sn matrix coefficients can be expressed as linear combinations of associated Legendre functions as relations (52), their recurrence relations result in other recurrence relations between the RC or RS coefficients of (52). However, achieving them is not easy and the detailed proofs are too lengthy to be given here. Their principle is as follows:

  1. Multiply both sides of equation (55) by |$\displaystyle\frac{{P_n^{m - k + 2p} (\cos \beta)}}{{\sin \beta }}$|Pnmk+2p(cosβ)sinβ to find RC recurrence or both sides of recurrence (56) by |$P_n^{m - k + 2q + 1} (\cos \beta)$|Pnmk+2q+1(cosβ) to find RS recurrence;

  2. Integrate both sides for β = 0 → π, which lives on the left side, in RC(n, m, k, μ) or RS(n, m, k, μ′) only through (A7);

  3. Transform the right sides by using (A9) and (A11) to achieve the sought recurrence relations.

N.B.: These recurrences are valid for mn − 1 but the explicit forms of RC and RS for m = n are already known, (62) and (69).

The relations between coefficients for the original and quasi-normalized Legendre functions are as follows:

\begin{equation} \begin{array}{l} R_{C_S } \left( {n,m,k,\mu = m - k + 2p} \right) = \\[10pt] R_C \left( {n,m,k,\mu = m - k + 2p} \right)\sqrt {\displaystyle\frac{1}{{2 - \delta _\mu ^0 }}\displaystyle\frac{{\left( {n + \mu } \right)!}}{{\left( {n - \mu } \right)!}}\displaystyle\frac{{2 - \delta _m^0 }}{{2 - \delta _k^0 }}\displaystyle\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}\displaystyle\frac{{\left( {n + k} \right)!}}{{\left( {n - k} \right)!}}} , \\[10pt] R_{S_S } \left( {n,m,k,\mu ' = m - k + 2q + 1} \right) = \\[10pt] R_S \left( {n,m,k,\mu ' = m - k + 2q + 1} \right)\sqrt {\displaystyle\frac{1}{{2 - \delta _{\mu '}^0 }}\displaystyle\frac{{\left( {n + \mu '} \right)!}}{{\left( {n - \mu '} \right)!}}\displaystyle\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}\displaystyle\frac{{\left( {n + k} \right)!}}{{\left( {n - k} \right)!}}} . \\ \end{array} \end{equation}
RCSn,m,k,μ=mk+2p=RCn,m,k,μ=mk+2p12δμ0n+μ!nμ!2δm02δk0nm!n+m!n+k!nk!,RSSn,m,k,μ=mk+2q+1=RSn,m,k,μ=mk+2q+112δμ0n+μ!nμ!nm!n+m!n+k!nk!.
(79)

Even if they are simple to code, the recurrence relations are rather impressive as displayed below, only for the quasi-normalized case:

\begin{equation*} \begin{array}{l} \displaystyle\frac{{\sqrt {\left( {n - m} \right)\left( {n + m} \right)} }}{\mu }R_{C_S } \left( {n,m,k,\left[ {\begin{array}{*{20}c} p \\[10pt] {\mu = m - k + 2p} \\ \end{array}} \right]} \right) = \\[10pt] - \left( {1 + \delta _k^1 } \right)\sqrt {\displaystyle\frac{{2 - \delta _{\mu - 1}^0 }}{{2 - \delta _\mu ^0 }}\displaystyle\frac{{2 - \delta _{k - 1}^0 }}{{2 - \delta _k^0 }}\displaystyle\frac{{\left( {n + k - 1} \right)\left( {n + k} \right)}}{{\left( {n + \mu - 1} \right)\left( {n + \mu } \right)}}} \times \\[10pt] \left[ \begin{array}{c} R_{C_S } \left( {n - 1,m,k - 1,\left[ {\begin{array}{*{20}c} {p - 1} \\[10pt] {\mu - 1} \\ \end{array}} \right]} \right) \\[10pt] - \sqrt {\displaystyle\frac{{2 - \delta _{\mu - 3}^0 }}{{2 - \delta _{\mu - 1}^0 }}\displaystyle\frac{{\left( {n - \mu {\rm + 1}} \right)\left( { + n - \mu {\rm + 2}} \right)}}{{\left( {n + \mu - 3} \right)\left( {n + \mu - 2} \right)}}} R_{C_S } \left( {n - 1,m,k - 1,\left[ {\begin{array}{*{20}c} {p - 2} \\[10pt] {\mu - 3} \\ \end{array}} \right]} \right) + \cdots \\ \end{array} \right] \\[10pt] + \sqrt {\left( {n - k} \right)\left( {n + k} \right)} \left[ \begin{array}{l} \displaystyle\frac{1}{\mu }\sqrt {\displaystyle\frac{{n - \mu }}{{n + \mu }}} R_{C_S } \left( {n - 1,m,k,\left[ {\begin{array}{*{20}c} p \\[10pt] \mu \\ \end{array}} \right]} \right) \\[10pt] + 2\sqrt {\displaystyle\frac{{2 - \delta _{\mu - 2}^0 }}{{2 - \delta _\mu ^0 }}\displaystyle\frac{{n - \mu + 1}}{{\left( {n + \mu - 2} \right)\left( {n + \mu - 1} \right)\left( {n + \mu } \right)}}} \\[10pt] \times R_{C_S } \left( {n - 1,m,k,\left[ {\begin{array}{*{20}c} {p - 1} \\[10pt] {\mu - 2} \\ \end{array}} \right]} \right) - \cdots \\ \end{array} \right] \\[10pt] \end{array} \end{equation*}
nmn+mμRCSn,m,k,pμ=mk+2p=1+δk12δμ102δμ02δk102δk0n+k1n+kn+μ1n+μ×RCSn1,m,k1,p1μ12δμ302δμ10nμ+1+nμ+2n+μ3n+μ2RCSn1,m,k1,p2μ3++nkn+k1μnμn+μRCSn1,m,k,pμ+22δμ202δμ0nμ+1n+μ2n+μ1n+μ×RCSn1,m,k,p1μ2
\begin{eqnarray} && + \sqrt {\displaystyle\frac{{2 - \delta _{\mu - 1}^0 }}{{2 - \delta _\mu ^0 }}\displaystyle\frac{{2 - \delta _{k + 1}^0 }}{{2 - \delta _k^0 }}\displaystyle\frac{{\left( {n - k - 1} \right)\left( {n - k} \right)}}{{\left( {n + \mu - 1} \right)\left( {n + \mu } \right)}}} \times \\ && \left[ \begin{array}{l} R_{C_S } \left( {n - 1,m,k + 1,\left[ {\begin{array}{*{20}c} p \\[10pt] {\mu - 1} \\ \nonumber \end{array}} \right]} \right) \\[10pt] - \sqrt {\displaystyle\frac{{2 - \delta _{\mu - 3}^0 }}{{2 - \delta _{\mu - 1}^0 }}\displaystyle\frac{{\left( {n - \mu {\rm + 1}} \right)\left( {n - \mu {\rm + 2}} \right)}}{{\left( {n + \mu - 3} \right)\left( {n + \mu - 2} \right)}}} R_{C_S } \left( {n - 1,m,k + 1,\left[ {\begin{array}{*{20}c} {p - 1} \\[10pt] {\mu - 3} \\ \end{array}} \right]} \right) + \cdots \\ \end{array} \right] \end{eqnarray}
+2δμ102δμ02δk+102δk0nk1nkn+μ1n+μ×RCSn1,m,k+1,pμ12δμ302δμ10nμ+1nμ+2n+μ3n+μ2RCSn1,m,k+1,p1μ3+
(80)
\begin{equation} \begin{array}{l} \displaystyle\frac{{\sqrt {\left( {n - m} \right)\left( {n + m} \right)} }}{{\mu '}}R_{S_S } \left( {n,m,k,\left[ {\begin{array}{*{20}c} q \\[10pt] {\mu ' = m - k + 2q + 1} \\ \end{array}} \right]} \right) = \\[10pt] - \sqrt {\displaystyle\frac{{2 - \delta _{\mu ' - 1}^0 }}{{2 - \delta _{\mu '}^0 }}\displaystyle\frac{{\left( {n + k - 1} \right)\left( {n + k} \right)}}{{\left( {n + \mu ' - 1} \right)\left( {n + \mu '} \right)}}} R_{S_S } \left( {n - 1,m,k - 1,\left[ {\begin{array}{*{20}c} {q - 1} \\[10pt] {\mu ' - 1} \\ \end{array}} \right]} \right) \\[10pt] + \sqrt {\displaystyle\frac{{2 - \delta _{\mu ' - 3}^0 }}{{2 - \delta _{\mu '}^0 }}\displaystyle\frac{{\left( {n + k - 1} \right)\left( {n + k} \right)\left( {n - \mu ' + 1} \right)\left( {n - \mu ' + 2} \right)}}{{\left( {n + \mu ' - 3} \right)\left( {n + \mu ' - 2} \right)\left( {n + \mu ' - 1} \right)\left( {n + \mu '} \right)}}} \\[10pt] \times R_{S_S } \left( {n - 1,m,k - 1,\left[ {\begin{array}{*{20}c} {q - 2} \\[10pt] {\mu ' - 3} \\ \end{array}} \right]} \right) + \cdots \\[10pt] + \sqrt {\left( {n - k} \right)\left( {n + k} \right)} \left[ \begin{array}{l} \displaystyle\frac{1}{{\mu '}}\sqrt {\displaystyle\frac{{n - \mu '}}{{n + \mu '}}} R_{S_S } \left( {n - 1,m,k,\left[ {\begin{array}{*{20}c} q \\[10pt] {\mu '} \\ \end{array}} \right]} \right) \\[10pt] + 2\sqrt {\displaystyle\frac{{2 - \delta _{\mu ' - 2}^0 }}{{2 - \delta _{\mu '}^0 }}\displaystyle\frac{{n - \mu ' + 1}}{{\left( {n + \mu ' - 2} \right)\left( {n + \mu ' - 1} \right)\left( {n + \mu '} \right)}}} \\[10pt] \times R_{S_S } \left( {n - 1,m,k,\left[ {\begin{array}{*{20}c} {q - 1} \\[10pt] {\mu ' - 2} \\ \end{array}} \right]} \right) - \cdots \\ \end{array} \right] \\[10pt] + {\kern 1pt} \sqrt {\displaystyle\frac{{2 - \delta _{\mu ' - 1}^0 }}{{2 - \delta _{\mu '}^0 }}\displaystyle\frac{{\left( {n - k - 1} \right)\left( {n - k} \right)}}{{\left( {n + \mu ' - 1} \right)\left( {n + \mu '} \right)}}} R_{S_S } \left( {n - 1,m,k + 1,\left[ {\begin{array}{*{20}c} q \\[10pt] {\mu ' - 1} \\ \end{array}} \right]} \right) \\[10pt] - \sqrt {\displaystyle\frac{{2 - \delta _{\mu ' - 3}^0 }}{{2 - \delta _{\mu '}^0 }}\displaystyle\frac{{\left( {n - k - 1} \right)\left( {n - k} \right)\left( {n - \mu ' + 1} \right)\left( {n - \mu ' + 2} \right)}}{{\left( {n + \mu ' - 3} \right)\left( {n + \mu ' - 2} \right)\left( {n + \mu ' - 1} \right)\left( {n + \mu '} \right)}}} \\[10pt] \times R_{S_S } \left( {n - 1,m,k + 1,\left[ {\begin{array}{*{20}c} {q - 1} \\[10pt] {\mu ' - 3} \\ \end{array}} \right]} \right) + \cdots \\ \end{array} \end{equation}
nmn+mμRSSn,m,k,qμ=mk+2q+1=2δμ102δμ0n+k1n+kn+μ1n+μRSSn1,m,k1,q1μ1+2δμ302δμ0n+k1n+knμ+1nμ+2n+μ3n+μ2n+μ1n+μ×RSSn1,m,k1,q2μ3++nkn+k1μnμn+μRSSn1,m,k,qμ+22δμ202δμ0nμ+1n+μ2n+μ1n+μ×RSSn1,m,k,q1μ2+2δμ102δμ0nk1nkn+μ1n+μRSSn1,m,k+1,qμ12δμ302δμ0nk1nknμ+1nμ+2n+μ3n+μ2n+μ1n+μ×RSSn1,m,k+1,q1μ3+
(81)

N.B.: The +⋅⋅⋅ must be understood in the context of the set of integrals (A9) and similarly the −⋅⋅⋅ in the context of integrals (A11) and the sequences stop when the integral no longer exists. Likewise, any RC or RS one index of which is out of its range must be considered as zero.

As mentioned in Section III A, rotation |$\beta = {\raise0.5ex\hbox{\scriptstyle \pi }\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{\scriptstyle 2}}$|β=π/2 plays a special role and it would be of interest to find specific recurrence relations for the matrix elements and eventually closed forms. These specific recurrences result directly from (64) and (71) or from (77) and (78) by setting sin β = 1 and cos β = 0:

\begin{equation} \begin{array}{l} \left( {1 + \delta _k^0 } \right)C_n^{m,k} + 2\left[ {n\left( {n + 1} \right) - 2m^2 - \left( {k + 2} \right)^2 } \right]C_n^{m,k + 2} \\[10pt] + \left( {n - k - 3} \right)\left( {n - k - 2} \right)\left( {n + k + 3} \right)\left( {n + k + 4} \right)C_n^{m,k + 4} = 0 \\ \end{array} \end{equation}
1+δk0Cnm,k+2nn+12m2k+22Cnm,k+2+nk3nk2n+k+3n+k+4Cnm,k+4=0
(82)
\begin{equation} \begin{array}{l} S_n^{m,k} + 2\left[ {n\left( {n + 1} \right) - 2m^2 - \left( {k + 2} \right)^2 } \right]S_n^{m,k + 2} \\[10pt] + \left( {n - k - 3} \right)\left( {n - k - 2} \right)\left( {n + k + 3} \right)\left( {n + k + 4} \right)S_n^{m,k + 4} = 0 \\ \end{array} \end{equation}
Snm,k+2nn+12m2k+22Snm,k+2+nk3nk2n+k+3n+k+4Snm,k+4=0
(83)
\begin{equation} \begin{array}{l} \sqrt {1 + \delta _k^0 } C_{Sn}^{m,k} + 2\displaystyle\frac{{n\left( {n + 1} \right) - 2m^2 - \left( {k + 2} \right)^2 }}{{\sqrt {\left( {n - k - 1} \right)\left( {n - k} \right)\left( {n + k + 1} \right)\left( {n + k + 2} \right)} }}C_{sn}^{m,k + 2} \\[10pt] + \sqrt {\displaystyle\frac{{\left( {n - k - 3} \right)\left( {n - k - 2} \right)\left( {n + k + 3} \right)\left( {n + k + 4} \right)}}{{\left( {n - k - 1} \right)\left( {n - k} \right)\left( {n + k + 1} \right)\left( {n + k + 2} \right)}}} C_{sn}^{m,k + 4} = 0 \\ \end{array} \end{equation}
1+δk0CSnm,k+2nn+12m2k+22nk1nkn+k+1n+k+2Csnm,k+2+nk3nk2n+k+3n+k+4nk1nkn+k+1n+k+2Csnm,k+4=0
(84)
\begin{equation} \begin{array}{l} S_{Sn}^{m,k} + 2\displaystyle\frac{{n\left( {n + 1} \right) - 2m^2 - \left( {k + 2} \right)^2 }}{{\sqrt {\left( {n - k - 1} \right)\left( {n - k} \right)\left( {n + k + 1} \right)\left( {n + k + 2} \right)} }}S_{Sn}^{m,k + 2} \\[10pt] + \sqrt {\displaystyle\frac{{\left( {n - k - 3} \right)\left( {n - k - 2} \right)\left( {n + k + 3} \right)\left( {n + k + 4} \right)}}{{\left( {n - k - 1} \right)\left( {n - k} \right)\left( {n + k + 1} \right)\left( {n + k + 2} \right)}}} S_{Sn}^{m,k + 4} = 0 \\ \end{array} \end{equation}
SSnm,k+2nn+12m2k+22nk1nkn+k+1n+k+2SSnm,k+2+nk3nk2n+k+3n+k+4nk1nkn+k+1n+k+2SSnm,k+4=0
(85)

As already mentioned, equations (82) and (83) are easier to handle for mathematical work while equations (84) and (85) are recommended for numerical computations, especially for high degree matrices. One could rely on the Section III D explicit formulas toobtain the necessary first terms but it is much simpler to use these recurrences backwards since then only the two last columns of both matrices are needed. The last column is obtained from (61) or (68) by using the symmetry relations (33):

\begin{equation} \begin{array}{l} C_n^{m,n} = \left( { - 1} \right)^n \displaystyle\frac{{1 + \left( { - 1} \right)^m }}{{2^n }}\displaystyle\frac{1}{{\left( {n - m} \right)!}} , \\[10pt] S_n^{m,n} = \left( { - 1} \right)^{n + 1} \displaystyle\frac{{1 - \left( { - 1} \right)^m }}{{2^{n - 1} }}\displaystyle\frac{1}{{\left( {n - m} \right)!}} . \\ \end{array} \end{equation}
Cnm,n=1n1+1m2n1nm!,Snm,n=1n+111m2n11nm!.
(86)

The second-to-last column is then deduced from crossed relations (37):

\begin{equation} \begin{array}{l} C_n^{m,n - 1} = 2m{\kern 1pt} S_n^{m,n} = \left( { - 1} \right)^{n + 1} \displaystyle\frac{{1 - \left( { - 1} \right)^m }}{{2^{n - 1} }}\displaystyle\frac{m}{{\left( {n - m} \right)!}} , \\[10pt] S_n^{m,n - 1} = 2m{\kern 1pt} {\kern 1pt} C_n^{m,n} = \left( { - 1} \right)^n \displaystyle\frac{{1 + \left( { - 1} \right)^m }}{{2^{n - 1} }}\displaystyle\frac{m}{{\left( {n - m} \right)!}} . \\ \end{array} \end{equation}
Cnm,n1=2mSnm,n=1n+111m2n1mnm!,Snm,n1=2mCnm,n=1n1+1m2n1mnm!.
(87)

Recurrences (82) and (83) lead directly to the two previous columns by setting k = n − 2 and k = n − 3 since only two terms are left in the recurrences:

\begin{equation} \begin{array}{l} \left( {1 + \delta _{n - 2}^0 } \right){\kern 1pt} {\kern 1pt} C_n^{m,n - 2} = \left( { - 1} \right)^n \displaystyle\frac{{1 + \left( { - 1} \right)^m }}{{2^{n - 1} }}\displaystyle\frac{{2m^2 - n}}{{\left( {n - m} \right)!}} , \\[10pt] S_n^{m,n - 2} = \left( { - 1} \right)^{n + 1} \displaystyle\frac{{1 - \left( { - 1} \right)^m }}{{2^{n - 1} }}\displaystyle\frac{{2m^2 - n}}{{\left( {n - m} \right)!}} . \\ \end{array} \end{equation}
1+δn20Cnm,n2=1n1+1m2n12m2nnm!,Snm,n2=1n+111m2n12m2nnm!.
(88)
\begin{equation} \hspace*{-6pt}\begin{array}{l} \left( {1 + \delta _{n - 3}^0 } \right)C_n^{m,n - 3} = 2\left( {2m^2 - 3n + 1} \right)C_n^{m,n - 1} = \left( { - 1} \right)^{n + 1} \displaystyle\frac{{1 - \left( { - 1} \right)^m }}{{2^{n - 2} }}\displaystyle\frac{{m\left( {2m^2 - 3n + 1} \right)}}{{\left( {n - m} \right)!}} , \\[10pt] S_n^{m,n - 3} = 2\left( {2m^2 - 3n + 1} \right)S_n^{m,n - 1} = \left( { - 1} \right)^n \displaystyle\frac{{1 + \left( { - 1} \right)^m }}{{2^{n - 2} }}\displaystyle\frac{{m\left( {2m^2 - 3n + 1} \right)}}{{\left( {n - m} \right)!}} . \\ \end{array} \end{equation}
1+δn30Cnm,n3=22m23n+1Cnm,n1=1n+111m2n2m2m23n+1nm!,Snm,n3=22m23n+1Snm,n1=1n1+1m2n2m2m23n+1nm!.
(89)

The last columns of the CSn or SSn matrices which are needed for using relations (84) or (85) backwards are obtained from that of Cn or Sn through relations (25).

Since (82) and (83) are single-index three-term recurrence relations the coefficients of which are polynomial functions of the index, they can be solved by classical techniques. These closed-form solutions are certainly of academic interest but are much less convenient for calculations than the recurrence relations themselves. One of them (m = 3, n odd, k odd) is given below as an example:

\begin{equation} \begin{array}{l} S_n^{3,k} = \left( { - 1} \right)^{\frac{{n - k}}{2}} \displaystyle\frac{{k\left[ {4k^2 + 2 - 3n\left( {n + 1} \right)} \right]}}{{2^{n - 1} }}\displaystyle\frac{{\left( {n - k} \right)!}}{{\left( {\displaystyle\frac{{n - k}}{2}} \right)!\left( {\displaystyle\frac{{n + k}}{2}} \right)!}} , \\[10pt] n = 2p + 1,\;p \ge 1,\;k = 2q + 1,\;0 \le q \le p . \\ \end{array} \end{equation}
Sn3,k=1nk2k4k2+23nn+12n1nk!nk2!n+k2!,n=2p+1,p1,k=2q+1,0qp.
(90)

N.B.: Complexity of these closed forms increases with m and nm, that is, when going toward the middle rows.

Numerical values for n = 9 are displayed below for a possible check of both explicit formulas and recurrence relations:

\begin{equation} \begin{array}{l} S_9^{3,1} = - \displaystyle\frac{{231}}{{16}},\;S_9^{3,3} = \displaystyle\frac{{29}}{{64}},\;S_9^{3,5} = - \displaystyle\frac{1}{{128}},\;S_9^{3,7} = \displaystyle\frac{1}{{10240}},\;S_9^{3,9} = \displaystyle\frac{1}{{184320}} , \\[10pt] S_{S9}^{3,1} = - \displaystyle\frac{{\sqrt {462} }}{{128}},\;S_{S9}^{3,3} = \displaystyle\frac{{29}}{{64}},\;S_{S9}^{3,5} = - \displaystyle\frac{{\sqrt {1365} }}{{64}},\;S_{S9}^{3,7} = \displaystyle\frac{{3\sqrt {273} }}{{128}},\;S_{S9}^{3,9} = \displaystyle\frac{{\sqrt {4641} }}{{128}} . \\ \end{array} \end{equation}
S93,1=23116,S93,3=2964,S93,5=1128,S93,7=110240,S93,9=1184320,SS93,1=462128,SS93,3=2964,SS93,5=136564,SS93,7=3273128,SS93,9=4641128.
(91)

To complete the picture, relations between CSn, SSn and the original Wigner d-matrices are established now, leading to explicit formulas for all their elements. Wigner d-matrices are defined as follows:

\begin{equation} \begin{array}{l} Y_n^m (\vartheta,\varphi) = \displaystyle\sum\limits_{k = - n}^n {d_{m,k}^n (\beta){\kern 1pt} {\kern 1pt} Y_n^k (\vartheta ',\varphi ')} , \\[10pt] d_{m,k}^n (\beta) = \sqrt {\displaystyle\frac{{\left( {n - m} \right)!\left( {n + m} \right)!}}{{\left( {n - k} \right)!\left( {n + k} \right)!}}} \\[10pt] \times \displaystyle\sum\limits_{s = \max (0,k - m)}^{\min (n - m,n + k)} {\left( { - 1} \right)^{m - k + s} \left( {\begin{array}{*{20}c} {n + k} \\[10pt] s \\ \end{array}} \right)\left( {\begin{array}{*{20}c} {n - k} \\[10pt] {m - k + s} \\ \end{array}} \right)\cos ^{2n - m + k - 2s} \displaystyle\frac{\beta }{2}\sin ^{m - k + 2s} \displaystyle\frac{\beta }{2}} . \\ \end{array} \end{equation}
Ynm(ϑ,φ)=k=nndm,kn(β)Ynk(ϑ,φ),dm,kn(β)=nm!n+m!nk!n+k!×s=max(0,km)min(nm,n+k)1mk+sn+ksnkmk+scos2nm+k2sβ2sinmk+2sβ2.
(92)

N.B.: Using binomial coefficients instead of the original Wigner notations with factorials as in ref. 3 or 4 makes the range of s easier to understand and leads to more elegant formulas.

Besides the well-known relations

\begin{equation} \begin{array}{l} d_{m,k}^n (\beta) = \left( { - 1} \right)^{m - k} d_{k,m}^n (\beta) , \\[10pt] d_{m,k}^n (\beta) = \left( { - 1} \right)^{m - k} d_{ - m, - k}^n (\beta) , \\ \end{array}\end{equation}
dm,kn(β)=1mkdk,mn(β),dm,kn(β)=1mkdm,kn(β),
(93)

another important relation is achieved by properly transforming the indices in (92):

\begin{equation} d_{m, - k}^n (\beta) = \left( { - 1} \right)^{2\left( {n - \left[ n \right]} \right)} \left( { - 1} \right)^{n - m} d_{m,k}^n (\pi - \beta) \end{equation}
dm,kn(β)=12nn1nmdm,kn(πβ)
(94)

N.B.: The factor ( − 1)2(n − [n]) is required only if n, m, k are half-integers (spin angular momentum).

For n, m, k integers, remarkably simple relations follow from (12), (14), (7), (92), and (94):

\begin{equation} \begin{array}{l} C_{Sn}^{m,k} = \displaystyle\frac{1}{{\sqrt {1 + \delta _m^0 } }}\displaystyle\frac{1}{{\sqrt {1 + \delta _k^0 } }}\left[ {\left( { - 1} \right)^{m + k} d_{m,k}^n (\beta) + \left( { - 1} \right)^n d_{m,k}^n (\pi - \beta)} \right] , \\[10pt] S_{Sn}^{m,k} = \left( { - 1} \right)^{m + k} d_{m,k}^n (\beta) - \left( { - 1} \right)^n d_{m,k}^n (\pi - \beta) . \\ \end{array} \end{equation}
CSnm,k=11+δm011+δk01m+kdm,kn(β)+1ndm,kn(πβ),SSnm,k=1m+kdm,kn(β)1ndm,kn(πβ).
(95)

Conversely, they give:

\begin{equation} \begin{array}{l} d_{m,k}^n = \left( { - 1} \right)^{m + k} \displaystyle\frac{1}{2}\left( {C_{Sn}^{m,k} + S_{Sn}^{m,k} } \right) ,\;1 \le m \le n,\;1 \le k \le n, \\[10pt] d_{m,0}^n = \left( { - 1} \right)^m \sqrt {\displaystyle\frac{{1 + \delta _m^0 }}{2}} {\kern 1pt} {\kern 1pt} C_{Sn}^{m,0} = \left( { - 1} \right)^m \sqrt {\displaystyle\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}} {\kern 1pt} {\kern 1pt} P_n^m (\cos \beta) ,\;0 \le m \le n. \\ \end{array} \end{equation}
dm,kn=1m+k12CSnm,k+SSnm,k,1mn,1kn,dm,0n=1m1+δm02CSnm,0=1mnm!n+m!Pnm(cosβ),0mn.
(96)

It becomes clear that any Wigner d-matrix element should be expressed as a sin β and cos β trigonometric polynomial. Indeed, the β dependent terms in (95) involve expressions called Ω below which are transformed through the following steps for switching from |${\raise0.5ex\hbox{\scriptstyle \beta }\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{\scriptstyle 2}}$|β/2 to β trigonometric functions:

\begin{eqnarray} \begin{array}{l} \Omega = \cos ^{2n - m + k - 2s} \displaystyle\frac{\beta }{2}\sin ^{m - k + 2s} \displaystyle\frac{\beta }{2} \pm \cos ^{2n - m + k - 2s} \displaystyle\frac{{\pi - \beta }}{2}\sin ^{m - k + 2s} \displaystyle\frac{{\pi - \beta }}{2} \\[10pt] = \cos ^{2n - m + k - 2s} \displaystyle\frac{\beta }{2}\sin ^{m - k + 2s} \displaystyle\frac{\beta }{2} \pm \sin ^{2n - m + k - 2s} \displaystyle\frac{\beta }{2}\cos ^{m - k + 2s} \displaystyle\frac{\beta }{2}, \\[10pt] p = m - k + 2s,\;\;q = 2n - p, \\[10pt] \Omega = \left[ {\begin{array}{*{20}c} {\sin ^p \displaystyle\frac{\beta }{2}\cos ^p \displaystyle\frac{\beta }{2}\left( {\cos ^{2\left( {n - p} \right)} \displaystyle\frac{\beta }{2} \pm \sin ^{2\left( {n - p} \right)} \displaystyle\frac{\beta }{2}} \right)} , & & {0 \le p \le n} \\[10pt] {\sin ^{2n - p} \displaystyle\frac{\beta }{2}\cos ^{2n - p} \displaystyle\frac{\beta }{2}\left( {\sin ^{2\left( {p - n} \right)} \displaystyle\frac{\beta }{2} \pm \cos ^{2\left( {p - n} \right)} \displaystyle\frac{\beta }{2}} \right)} , & & {n \le p \le 2n} \nonumber \\ \end{array}} \right], \\[24pt] \cos ^2 \displaystyle\frac{\beta }{2} = \displaystyle\frac{1}{2}\left( {1 + \cos \beta } \right),\;\;\sin ^2 \displaystyle\frac{\beta }{2} = \displaystyle\frac{1}{2}\left( {1 - \cos \beta } \right), \\[24pt] \Omega = \left[ {\begin{array}{*{20}c} {\displaystyle\frac{1}{{2^n }}\sin ^p \beta \left[ {\left( {1 + \cos \beta } \right)^{n - p} \pm \left( {1 - \cos \beta } \right)^{n - p} } \right]} , & & {0 \le p \le n} \\[10pt] {\displaystyle\frac{1}{{2^n }}\sin ^{2n - p} \beta \left[ {\left( {1 - \cos \beta } \right)^{p - n} \pm \left( {1 + \cos \beta } \right)^{p - n} } \right]} , & & {n \le p \le 2n} \\ \end{array}} \right] . \\[-1.3pc] \end{array} \\ \end{eqnarray}
Ω=cos2nm+k2sβ2sinmk+2sβ2±cos2nm+k2sπβ2sinmk+2sπβ2=cos2nm+k2sβ2sinmk+2sβ2±sin2nm+k2sβ2cosmk+2sβ2,p=mk+2s,q=2np,Ω=sinpβ2cospβ2cos2npβ2±sin2npβ2,0pnsin2npβ2cos2npβ2sin2pnβ2±cos2pnβ2,np2n,cos2β2=121+cosβ,sin2β2=121cosβ,Ω=12nsinpβ1+cosβnp±1cosβnp,0pn12nsin2npβ1cosβpn±1+cosβpn,np2n.
(97)

It is then possible to find the following closed form of a Wigner d-matrix element:

\begin{equation} \begin{array}{l} d_{m,k}^n (\beta) = \displaystyle\frac{1}{{2^k \left( {k - m} \right)!}}\sqrt {\displaystyle\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}\displaystyle\frac{{\left( {n + k} \right)!}}{{\left( {n - k} \right)!}}} \sin ^{k - m} \beta \left( {1 + \cos \beta } \right)^m \\[10pt] \times F\left( { - n + k,n + k + 1;k - m + 1;\displaystyle\frac{{1 - \cos \beta }}{2}} \right) ,\;\;0 \le m \le k \le n , \\ \end{array} \end{equation}
dm,kn(β)=12kkm!nm!n+m!n+k!nk!sinkmβ1+cosβm×Fn+k,n+k+1;km+1;1cosβ2,0mkn,
(98)

whereF(a, b; c; z) is a hypergeometric function as defined in ref. 9 or 11.

N.B.: For 0 ⩽ k < m use (93) and for negative values of k use (94) before (98). This formula is also valid for n, m, k half-integers (spin angular momentum).

This hypergeometric function can be written as a Jacobi polynomial by using formula 8.962.1 of ref. 10:

\begin{equation} \begin{array}{l} d_{m,k}^n (\beta) = \displaystyle\frac{1}{{2^k }}\sqrt {\displaystyle\frac{{\left( {n - k} \right)!}}{{\left( {n - m} \right)!}}\displaystyle\frac{{\left( {n + k} \right)!}}{{\left( {n + m} \right)!}}} \sin ^{k - m} \beta \left( {1 + \cos \beta } \right)^m P_{n - k}^{\left( {k - m,k + m} \right)} (\cos \beta) , \\[10pt] 0 \le m \le k \le n . \\ \end{array} \end{equation}
dm,kn(β)=12knk!nm!n+k!n+m!sinkmβ1+cosβmPnkkm,k+m(cosβ),0mkn.
(99)

This last form is equivalent to formula (3.74) of ref. 15 without the complexity of considering a table of different cases since only one is needed (0 ⩽ mkn) through (93) and especially (94) not mentioned in this reference.

New forms of the CSn and Ssn matrix elements are then obtained through (95) but naturally, such formulas should be used only for calculating single coefficients and recurrence relations should be preferred for calculating the whole matrices:

\begin{equation} \begin{array}{l} C_{Sn}^{m,k} = \displaystyle\frac{1}{{\sqrt {1 + \delta _m^0 } }}\displaystyle\frac{1}{{\sqrt {1 + \delta _k^0 } }}\displaystyle\frac{1}{{2^k }}\sqrt {\displaystyle\frac{{\left( {n - k} \right)!}}{{\left( {n - m} \right)!}}\displaystyle\frac{{\left( {n + k} \right)!}}{{\left( {n + m} \right)!}}} \sin ^{k - m} \beta \\[10pt] \times \left[ {\left( { - 1} \right)^{m + k} \left( {1 + \cos \beta } \right)^m P_{n - k}^{\left( {k - m,k + m} \right)} (\cos \beta) + \left( { - 1} \right)^n \left( {1 - \cos \beta } \right)^m P_{n - k}^{\left( {k - m,k + m} \right)} ( - \cos \beta)} \right] , \\[10pt] S_{Sn}^{m,k} = \displaystyle\frac{1}{{2^k }}\sqrt {\displaystyle\frac{{\left( {n - k} \right)!}}{{\left( {n - m} \right)!}}\displaystyle\frac{{\left( {n + k} \right)!}}{{\left( {n + m} \right)!}}} \sin ^{k - m} \beta \\[10pt] \times \left[ {\left( { - 1} \right)^{m + k} \left( {1 + \cos \beta } \right)^m P_{n - k}^{\left( {k - m,k + m} \right)} (\cos \beta) - \left( { - 1} \right)^n \left( {1 - \cos \beta } \right)^m P_{n - k}^{\left( {k - m,k + m} \right)} ( - \cos \beta)} \right] , \\[10pt] 0 \le m \le k \le n . \\ \end{array} \end{equation}
CSnm,k=11+δm011+δk012knk!nm!n+k!n+m!sinkmβ×1m+k1+cosβmPnkkm,k+m(cosβ)+1n1cosβmPnkkm,k+m(cosβ),SSnm,k=12knk!nm!n+k!n+m!sinkmβ×1m+k1+cosβmPnkkm,k+m(cosβ)1n1cosβmPnkkm,k+m(cosβ),0mkn.
(100)

N.B.: Elements of the lower triangle of these matrices are obtained through (32).

In the special case |$\beta = {\raise0.5ex\hbox{\scriptstyle \pi }\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{\scriptstyle 2}}$|β=π/2, (92) and (95) readily give:

\begin{equation} \begin{array}{l} C_n^{m,k} = \displaystyle\frac{1}{{1 + \delta _m^0 }}\left[ {1 + \left( { - 1} \right)^{n + m + k} } \right]U_n^{m,k} , \\[10pt] S_n^{m,k} = \left[ {1 - \left( { - 1} \right)^{n + m + k} } \right]U_n^{m,k} , \\[10pt] U_n^{m,k} = \displaystyle\frac{1}{{2^n }}\displaystyle\frac{{\left( {n + m} \right)!}}{{\left( {n - k} \right)!}}\displaystyle\sum\limits_{s = \max (0,k - m)}^{n - m} {\left( { - 1} \right)^s C_{n + k}^s C_{n - k}^{m - k + s} } . \\ \end{array} \end{equation}
Cnm,k=11+δm01+1n+m+kUnm,k,Snm,k=11n+m+kUnm,k,Unm,k=12nn+m!nk!s=max(0,km)nm1sCn+ksCnkmk+s.
(101)

This latter sum can be expressed as a hypergeometric function in two different ways and the lower triangle of the Un matrix reads:

\begin{equation} \begin{array}{l} U_n^{m,k \ge m} = \left( { - 1} \right)^{k - m} \displaystyle\frac{1}{{2^k \left( {k - m} \right)!}}F\left( { - n + k,n + k + 1;k - m + 1;\displaystyle\frac{1}{2}} \right) , \\[10pt] = \left( { - 1} \right)^{k - m} \displaystyle\frac{1}{{2^n \left( {k - m} \right)!}}F\left( { - n + k, - n - m;k - m + 1; - 1} \right) . \\ \end{array} \end{equation}
Unm,km=1km12kkm!Fn+k,n+k+1;km+1;12,=1km12nkm!Fn+k,nm;km+1;1.
(102)

The upper triangle follows from

\begin{equation} U_n^{k,m \le k} = \left( { - 1} \right)^{k - m} \displaystyle\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}\displaystyle\frac{{\left( {n + k} \right)!}}{{\left( {n - k} \right)!}}U_n^{m,k \ge m} \end{equation}
Unk,mk=1kmnm!n+m!n+k!nk!Unm,km
(103)

One can for instance check that both (90), (101), and (102) give |$S_9^{3,5} = - {\raise0.5ex\hbox{\scriptstyle 1}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{\scriptstyle {128}}}$|S93,5=1/128.

There are many ways to transform SH under rotation. They all attempt to build rotation matrices either by explicit formulas for their elements or through recurrence relations between them. The former are mainly of academic interest and for calculating some particular elements. The latter must be preferred when whole matrices are needed and for an accurate and fast numerical evaluation, raising the important question of their computational stability and efficiency. This paper establishes an extended set of results addressing all aspects of the problem and focusing on the most difficult case of RSH.

These results throw fresh light on the well-established case of CSH and on the Wigner d-matrix elements closed forms.

G.A. is indebted to D. Sakellariou for drawing his attention to this problem and to A. Chancé for carefully reading the manuscript and for his stimulating comments.

1. Relations between associated Legendre functions

There are a great number of functional relations between the |$P_n^m (\cos \vartheta)$|Pnm(cosϑ) s and their derivatives. Only those entering the various proofs of this work are recalled here. For the sake of simplicity, the arguments (cos ϑ) are omitted and any |$P_\nu ^\mu $|Pνμ which does not respect 0 ⩽ μ ⩽ ν must be considered as zero.

\begin{equation} \left( {2n + 1} \right)\cos \vartheta P_n^m = \left( {n - m + 1} \right)P_{n + 1}^m + \left( {n + m} \right)P_{n - 1}^m . \end{equation}
2n+1cosϑPnm=nm+1Pn+1m+n+mPn1m.
(A1)
\begin{equation} \left( {2n + 1} \right)\sin \vartheta P_n^m = P_{n + 1}^{m + 1} - P_{n - 1}^{m + 1} . \end{equation}
2n+1sinϑPnm=Pn+1m+1Pn1m+1.
(A2)
\begin{equation} \left( {2n + 1} \right)\sin \vartheta P_n^m = \left( {n + m} \right)\left( {n + m - 1} \right)P_{n - 1}^{m - 1} - \left( {n - m + 1} \right)\left( {n - m + 2} \right)P_{n + 1}^{m - 1}. \end{equation}
2n+1sinϑPnm=n+mn+m1Pn1m1nm+1nm+2Pn+1m1.
(A3)
\begin{equation} \begin{array}{l} 2\displaystyle\frac{{\cos \vartheta }}{{\sin \vartheta }}P_n^m = \displaystyle\frac{{\left( {n - m + 1} \right)\left( {n + m} \right)}}{m}P_n^{m - 1} + \displaystyle\frac{1}{m}P_n^{m + 1} , \\[10pt] \left( {2\displaystyle\frac{{\cos \vartheta }}{{\sin \vartheta }}} \right)^2 P_n^m = \displaystyle\frac{{\left( {n - m + 2} \right)!\left( {n + m} \right)!}}{{\left( {n - m} \right)!\left( {n + m - 2} \right)!}}\displaystyle\frac{{\left( {m - 2} \right)!}}{{m!}}P_n^{m - 2} \\[10pt] + 2\displaystyle\frac{{n^2 - m^2 + n + 1}}{{\left( {m - 1} \right)\left( {m + 1} \right)}}P_n^m + \displaystyle\frac{{\left( {m - 1} \right)!}}{{\left( {m + 1} \right)!}}P_n^{m + 2} , \\[10pt] \left( {2\displaystyle\frac{{\cos \vartheta }}{{\sin \vartheta }}} \right)^3 P_n^m = \displaystyle\frac{{\left( {n - m + 3} \right)!\left( {n + m} \right)!}}{{\left( {n - m} \right)!\left( {n + m - 3} \right)!}}\displaystyle\frac{{\left( {m - 3} \right)!}}{{m!}}P_n^{m - 3} , \\[10pt] + 3\displaystyle\frac{{n^2 - m^2 + n + m + 2}}{{\left( {m - 2} \right)\left( {m + 1} \right)}}\displaystyle\frac{{\left( {n - m + 1} \right)\left( {n + m} \right)}}{m}P_n^{m - 1} \\[10pt] + 3\displaystyle\frac{{n^2 - m^2 + n - m + 2}}{{\left( {m - 1} \right)\left( {m + 2} \right)}}\displaystyle\frac{1}{m}P_n^{m + 1} + \displaystyle\frac{{\left( {m - 1} \right)!}}{{\left( {m + 2} \right)!}}P_n^{m + 3} . \\[10pt] \cdots \\ \end{array} \end{equation}
2cosϑsinϑPnm=nm+1n+mmPnm1+1mPnm+1,2cosϑsinϑ2Pnm=nm+2!n+m!nm!n+m2!m2!m!Pnm2+2n2m2+n+1m1m+1Pnm+m1!m+1!Pnm+2,2cosϑsinϑ3Pnm=nm+3!n+m!nm!n+m3!m3!m!Pnm3,+3n2m2+n+m+2m2m+1nm+1n+mmPnm1+3n2m2+nm+2m1m+21mPnm+1+m1!m+2!Pnm+3.
(A4)

Each next relation of this set is obtained by successive application of the first (A4) equation to the right part of the previous relation in the set.

\begin{equation} \begin{array}{l} \displaystyle\frac{{dP_n^m (\cos \vartheta)}}{{d\vartheta }} = m\displaystyle\frac{{\cos \vartheta }}{{\sin \vartheta }} - P_n^{m + 1} (\cos \vartheta) , \\[10pt] \displaystyle\frac{{dP_n^m (\cos \vartheta)}}{{d\vartheta }} = - m\displaystyle\frac{{\cos \vartheta }}{{\sin \vartheta }} + \left( {n - m + 1} \right)\left( {n + m} \right)P_n^{m - 1} (\cos \vartheta) , \\[10pt] 2\displaystyle\frac{{dP_n^m }}{{d\vartheta }} = \left( {n - m + 1} \right)\left( {n + m} \right)P_n^{m - 1} - P_n^{m + 1} . \\ \end{array} \end{equation}
dPnm(cosϑ)dϑ=mcosϑsinϑPnm+1(cosϑ),dPnm(cosϑ)dϑ=mcosϑsinϑ+nm+1n+mPnm1(cosϑ),2dPnmdϑ=nm+1n+mPnm1Pnm+1.
(A5)
2. Integrals of products of associated Legendre functions

It seems that only two such integrals can be found in classical references like 9–11:

\begin{equation} \int_0^\pi {P_n^m (\cos \vartheta)P_\nu ^m (\cos \vartheta)\sin \vartheta d\vartheta } = \displaystyle\frac{2}{{2n + 1}}\displaystyle\frac{{\left( {n + m} \right)!}}{{\left( {n - m} \right)!}}\delta _n^\nu = \Phi _n^m \delta _n^\nu. \end{equation}
0πPnm(cosϑ)Pνm(cosϑ)sinϑdϑ=22n+1n+m!nm!δnν=Φnmδnν.
(A6)
\begin{equation} \int_0^\pi {P_n^m (\cos \vartheta)P_n^\mu (\cos \vartheta)\displaystyle\frac{{d\vartheta }}{{\sin \vartheta }}} = \displaystyle\frac{1}{m}\displaystyle\frac{{\left( {n + m} \right)!}}{{\left( {n - m} \right)!}}\delta _m^\mu = \Psi _n^m \delta _m^\mu. \end{equation}
0πPnm(cosϑ)Pnμ(cosϑ)dϑsinϑ=1mn+m!nm!δmμ=Ψnmδmμ.
(A7)

A first set of “new” integrals is as follows:

\begin{equation} \Sigma _n^{m,\mu } = \int_0^\pi {P_n^m (\cos \vartheta)P_{n - 1}^\mu (\cos \vartheta)d\vartheta } ,\;1 \le m \le n,\;1 \le \mu \le n - 1. \end{equation}
Σnm,μ=0πPnm(cosϑ)Pn1μ(cosϑ)dϑ,1mn,1μn1.
(A8)

It can be shown, by using particular equations (A5), (A2), and (A6), that these integrals are zero except if μ = m + 1, m + 3, … ⩾ 1, in which case:

\begin{equation} \Sigma _n^{m,m - 2p + 1} = \left( { - 1} \right)^{p + 1} 2\displaystyle\frac{{\left( {n + m - 2p} \right)!}}{{\left( {n - m} \right)!}} ,\;p = 1,2, \ldots \le \left[ {\displaystyle\frac{m}{2}} \right] . \end{equation}
Σnm,m2p+1=1p+12n+m2p!nm!,p=1,2,...m2.
(A9)

A second set is as follows:

\begin{equation} \Sigma_n^{\prime{m,\mu }} = \int_0^\pi {P_n^m (\cos \vartheta)P_{n - 1}^\mu (\cos \vartheta)\displaystyle\frac{{\cos \vartheta }}{{\sin \vartheta }}d\vartheta } ,\;1 \le m \le n,\;1 \le \mu \le n - 1. \end{equation}
Σnm,μ=0πPnm(cosϑ)Pn1μ(cosϑ)cosϑsinϑdϑ,1mn,1μn1.
(A10)

Using equation (A4) brings us back to the previous series with the result that these integrals are zero except whereμ = m, m − 2, m − 4, … ⩾ 1, in which case:

\begin{equation} \begin{array}{l} \Sigma_n^{\prime{m,m}} = \displaystyle\frac{1}{m}\displaystyle\frac{{\left( {n + m - 1} \right)!}}{{\left( {n - m - 1} \right)!}}, \\[10pt] \Sigma_n^{\prime{m,m - 2p}} = \left( { - 1} \right)^{p + 1} 2\displaystyle\frac{{\left( {n + m - 2p - 1} \right)!}}{{\left( {n - m} \right)!}} ,\;p = 1,2, \ldots \le \left[ {\displaystyle\frac{{m - 1}}{2}} \right]. \\ \end{array} \end{equation}
Σnm,m=1mn+m1!nm1!,Σnm,m2p=1p+12n+m2p1!nm!,p=1,2,...m12.
(A11)

Similar other “new” sets can be generated by the same techniques for the following integrals:

\begin{equation} \begin{array}{l} \displaystyle\int_0^\pi {P_n^m (\cos \vartheta)P_\nu ^\mu (\cos \vartheta)d\vartheta }, \\[18pt] \displaystyle\int_0^\pi {P_n^m (\cos \vartheta)P_\nu ^\mu (\cos \vartheta)\displaystyle\frac{{\cos \vartheta }}{{\sin \vartheta }}d\vartheta }, \\[18pt] \displaystyle 1 \le m \le n,\;1 \le \mu \le \nu , \;\nu = n - 2,\;n - 3, \ldots \ge 1. \\ \end{array} \end{equation}
0πPnm(cosϑ)Pνμ(cosϑ)dϑ,0πPnm(cosϑ)Pνμ(cosϑ)cosϑsinϑdϑ,1mn,1μν,ν=n2,n3,...1.
(A12)
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