A series of the form $\u2211\u2113=0\u221ec(\kappa ,\u2113)M\kappa ,\u2113+1/2(r0)W\kappa ,\u2113+1/2(r)P\u2113(cos(\gamma ))$ is evaluated explicitly where *c*(*κ*, *ℓ*) are suitable complex coefficients, *M*_{κ,μ} and *W*_{κ,μ} are the Whittaker functions, *P*_{ℓ} are the Legendre polynomials, *r*_{0} < *r* are radial variables, *γ* is an angle and *κ* is a complex parameter. The sum depends, as far as the radial variables and the angle are concerned, on their combinations *r* + *r*_{0} and $(r2+r02\u22122rr0\u2061cos(\gamma ))1/2$. This addition formula generalizes in some respect Gegenbauer’s Addition Theorem and follows rather straightforwardly from some already known results, particularly from Hostler’s formula for Coulomb Green’s function. In addition, several complementary summation formulas are derived. They suggest that a further extension of this addition formula may be possible.

## I. INTRODUCTION

Green’s function of a Hamiltonian is an important object in quantum physics as it contains, in principal, all information about the respective physical system. Particularly the set of singular points of Green’s function coincides with the spectrum. Green’s function is in fact the integral kernel of the resolvent of the Hamiltonian which is regarded as an integral operator. Sometimes, however, the integral kernel should be interpreted in the distributional sense. Green’s function is also closely related to the heat kernel or to the propagator. Namely, Green’s function is the Laplace transform of the heat kernel.

Green’s function can be explicitly expressed in a compact form for some quantum systems which are usually distinguished by their symmetry properties. As a rule, such systems frequently enjoy rotational symmetry. If so, this also opens the way to an alternative construction of Green’s function based on the method of separation of variables. The problem then effectively reduces to a one-dimensional one. Finally one deals with a positive second-order ordinary differential operator of Sturm-Liouville type though on the half-line rather than on a bounded interval. This is a substantial simplification since the construction of Green’s function for a Sturm-Liouville operator is commonly known and, in fact, this is a text-book matter. Thus in distinguished solvable cases the radial part of Green’s function can be expressed in terms of appropriate special functions. The full Hamiltonian depending on both radial and angle variables is then expressed as a sum with the summation running over eigenmodes of the spherical part of the Hamiltonian. Equaling the compact form of Green’s function to the sum obtained via the method of separation of variables leads to an addition formula for the involved special functions.

This procedure can be successfully applied to the Hamiltonian of the hydrogen atom, resulting in an addition formula for the Whittaker functions *M*_{κ,μ} and *W*_{κ,μ}. The formula turns out to be a generalization of Gegenbauer’s Addition Theorem in some respect. To the best of author’s knowledge, this possibility has not been exploited yet and still remains overlooked. And this is despite the fact that a compact formula for Coulomb Green’s function has been derived by Hostler rather long time ago in Ref. 10. As a companion of the addition formula for the Whittaker functions we further derive another addition formula concerning the Laguerre polynomials. There already exists a well-known addition formula for the Laguerre polynomials but that one reported here is completely different.

From the mathematical point of view, the formula for the Whittaker functions cannot be considered fully satisfactory, however, as the resulting sum involves only the Whittaker functions with the parameter *μ* = 1/2. A more general formula for arbitrary parameters *κ* and *μ* seems to be lacking. Nevertheless here we present some partial results in this direction which indicate that the derived formula could be further generalized. In addition to the parameter *κ*, with *μ* being restricted to the values 1/2 modulo integers, the formula depends on the radial variables *r*_{0} and *r* and on an angle *γ*. In the particular case *γ* = *π* we show that there exists an addition formula admitting general values of both *κ* and *μ*. Further we derive a summation formula for the Whittaker functions *W*_{κ,μ} only and another one for the Whittaker functions *M*_{κ,μ}. Again, in both cases, the parameters *κ* and *μ* can take arbitrary values.

The paper is organized as follows. In Sec. II we summarize some known formulas and results which are essential for the solution of our problem. The main result of the paper, namely a derivation of an addition formula for the Whittaker functions, is the content of Sec. III. Section IV is devoted to an addition formula for the Laguerre polynomials. Finally, Sec. V contains some complementary results suggesting that further generalizations could be possible, as discussed above.

## II. PRELIMINARIES

^{1,9}The Whittaker functions are defined in terms of the confluent hypergeometric functions,

*K*

_{ν},

*I*

_{ν}are particular cases of the Whittaker functions,’

*W*

_{n+(α+1)/2,α/2}and

*M*

_{n+(α+1)/2,α/2}are linearly dependent,

*n*∈

**Z**

_{+},

*n*≥ 1,

*μ*→

*∞*in $C$, Re(

*μ*) > 0, and $\kappa \u2208C$ is fixed,

*z*in a bounded region in $C$, and

*x*[one can also consult (Ref. 13, Chap. 7, Sec. 11.1) or Ref. 5].

*p*(

*x*) > 0 and

*q*(

*x*) ≥ 0 are sufficiently regular functions. At the finite endpoints one imposes mixed boundary conditions or, if the interval is unbounded, one requires functions from the domain of

*L*to be square integrable on a neighborhood of infinity. One assumes that

*L*with properly chosen boundary conditions is positive definite. To describe Green’s function of

*L*one finds two nontrivial solutions

*v*

_{0},

*v*

_{1}of the differential equation

*v*

_{0}satisfies the boundary condition at the left endpoint (or minus infinity) only while

*v*

_{1}satisfies the boundary condition at the right endpoint (or plus infinity) only. Then

*L*

^{−1}is an integral operator with the integral kernel

*v*

_{0}and

*v*

_{1}. Note that

*p*(

*x*)

*w*(

*x*) is in fact a constant function. Here and in the sequel

*ϑ*denotes the Heaviside step function.

^{2}+

*k*

^{2},

*k*> 0, in $R2$. Naturally, the partial differential operator is expressed in polar coordinates. Using the method of separation of variables one finds that Green’s function can be written in the form

*f*

_{n}(

*r*,

*r*

_{0}) can be obtained as solutions of the Sturm-Liouville problem in the radial variable, as described above, and we get

*φ*

_{0}= 0 and, by comparison, we obtain an addition formula for the modified Bessel functions,

*r*= |

**|, $r0=r0$)**

*r*^{2}+

*k*

^{2},

*k*> 0, in $R3$. Doing so one obtains a particular case of Gegenbauer’s Addition Theorem. We are not going to discuss this case, however. Instead, in Sec. III, we will focus on the operator −∇

^{2}−

*g*/|

**| +**

*r**k*

^{2},

*g*> 0 and

*k*> 0, in $R3$. Nevertheless let us recall what Gegenbauer’s theorem claims if specialized to the modified Bessel functions. For 0 ≤

*r*

_{0}<

*r*and $\nu \u2208R$,

*R*defined in Ref. 11, see (Ref. 16, Sec. II.4) [and also Ref. 1 (Eq. 9.1.80)]. Here $Cn(\nu )(z)$ are the Gegenbauer polynomials.

*m*| ≤

*ℓ*,

*θ*∈ [ 0,

*π*],

*φ*∈ [ 0, 2

*π*] are coordinates on the unit sphere

*S*

^{2}and $P\u2113m(z)$ is the associated Legendre polynomial. We have

*L*

^{2}(

*S*

^{2}, dΩ) and

*γ*) =

**·**

*n*

*n*_{0}, $n\u2254sin(\theta )cos(\phi ),sin(\theta )sin(\phi ),cos(\theta )$ and

*n*_{0}is defined similarly. $P\u2113(z)\u2261P\u21130(z)$ are the Legendre polynomials. Referring to (15) and (16), Eq. (17) means that

^{8}and a dozen different proofs of it have been provided, some of them quite intricate.

^{12}A straightforward derivation, as encountered in physical literature, is based on symmetry considerations and can be briefly rephrased as follows. Observe that

*L*

^{2}(

*S*

^{2},

*d*Ω) onto the eigenspace of minus the Laplace-Beltrami operator on

*S*

^{2}(denoted as $\u2212\Delta S2$) corresponding to the eigenvalue

*ℓ*(

*ℓ*+ 1). Thus we have (the differential operator acts in variables

*θ*and

*φ*)

**,**

*n*

*n*_{0}∈

*S*

^{2}only which in turn is a function of

**·**

*n*

*n*_{0}= cos(

*γ*). Writing $P\u0303\u2113(n,n0)=f(n\u22c5n0)$ Eq. (18) reduces to the ordinary second-order differential equation

*f*(

*z*) =

*c*

_{1}

*P*

_{ℓ}(

*z*) +

*c*

_{2}

*Q*

_{ℓ}(

*z*). Here

*Q*

_{ℓ}(

*z*) is the Legendre function of the second kind which is another independent solutions of the differential equation and is known to be singular for

*z*= ±1. Therefore $P\u0303\u2113(n,n0)=c1P\u2113(n\u22c5n0)$. The multiplicative constant is easily found to be

*c*

_{1}= (2

*ℓ*+ 1)/(4

*π*) when letting

**=**

*n*

*n*_{0}= (0, 0, 1) and taking into account that

*P*

_{ℓ}(1) = 1 and $P\u2113m(1)=0$ for

*m*≠ 0.

## III. THE HYDROGEN ATOM AND AN ADDITION FORMULA FOR THE WHITTAKER FUNCTIONS

*r*> 0,

*θ*∈ [0,

*π*],

*φ*∈ [0, 2

*π*], we denote

*H*= −∇

^{2}−

*g*/

*r*,

*g*> 0, and we wish to apply the procedure leading to an addition formula, as described in Sec. I, to the operator

*H*coincides with the positive real half-line, the discrete spectrum consists of eigenvalues

*E*

_{n}= −

*g*

^{2}/4

*n*

^{2}, $n\u2208N$, the multiplicity of

*E*

_{n}equals

*n*

^{2}. The corresponding normalized eigenfunctions are

*m*| ≤

*ℓ*≤

*n*− 1.

^{14,15}We have

^{16}this equation can be further simplified,

^{10},

*r*

_{0}<

*r*,

*g*= 2

*kκ*and rescaling

*r*→

*r*/(2

*k*),

*r*

_{0}→

*r*

_{0}/(2

*k*) we get an addition formula for the Whittaker functions. Moreover, the parameter

*κ*can be extended to complex values by analyticity.

*For*0 ≤

*r*

_{0}<

*r*

*,*$\kappa \u2208C\N$

*and*$\gamma \u2208R$

*,*

*where*

*R*=

*R*(

*γ*)

*, see Ref. 11,*

*and*

*κ*= 1 we have

*γ*= 0 (hence

*R*=

*r*−

*r*

_{0},

*x*= 2

*r*,

*y*= 2

*r*

_{0}) we have

*γ*=

*π*[hence

*R*=

*r*+

*r*

_{0},

*x*= 2(

*r*+

*r*

_{0}),

*y*= 0] we have

*z*= 0 on the LHS is removable). It can be proven by exploring the asymptotic behavior of both sides of (25), as

*r*→

*∞*. We have, in virtue of (7),

*z*instead of

*r*

_{0}, we see that (26) holds for

*z*> 0. But the asymptotic behavior of

*M*

_{κ,ℓ+1/2}(

*z*) for

*ℓ*large, as recalled in Eq. (8), which is locally uniform in

*z*implies that the LHS is an entire function of

*z*. Since the same is true for the RHS we conclude that (26) must hold for all $z\u2208C$.

*κ*= 0 (corresponding to

*g*= 0) in (22) we obtain a simplified equation

*ν*= 1/2. To derive (27) from (22) we have used (3) and (4) and also the equation

*κ*= 0 we get

## IV. AN ADDITION FORMULA FOR THE LAGUERRE POLYNOMIALS

*For*$n\u2208N$

*,*0 ≤

*r*

_{0}<

*r*

*and*$\gamma \u2208R$

*,*

*where*

*x*

*,*

*y*

*are defined in (23), with*

*R*=

*R*(

*γ*)

*, see Eq. (12).*

*z*=

*E*

_{n}equals

*E*

_{n}. $Pn$ is an integral operator with the integral kernel

*r*→ (

*n*/

*g*)

*r*,

*r*

_{0}→ (

*n*/

*g*)

*r*

_{0}, we get from (31) and (32)

*γ*= 0 we have

*γ*=

*π*we get

*L*

_{n}(0) = 1 [and $Ln1(0)=n+1$].

After the shift *n* → *n* + 1 one observes, in these two particular cases, that both sides are symmetric polynomials in *r* and *r*_{0}, and therefore *r*, *r*_{0} can be replaced by arbitrary complex variables.

*For every*$n\u2208Z+$

*and all*$u,v\u2208C$

*,*

*and*

^{3}Using expression for the residue of Green’s function, see (29) and (32), we get, for

*n*≥ 1,

## V. SOME COMPLEMENTARY SUMMATION FORMULAS

The following proposition presents a summation formula for the Whittaker functions *W*_{κ,μ} and is a straightforward corollary of Theorem 8 below which in turn generalizes the addition formula (25). Nevertheless this proposition should be proven independently because, conversely, it is used in the Proof of Theorem 8.

*For every*$n\u2208N$

*,*

*r*> 0

*,*$\kappa ,\mu \u2208C$

*,*2

*μ*≠ −1, −2, −3, …

*,*

*n*. For

*n*= 0 the equation is trivial. For

*n*= 1 this is a well known identity [for instance, this is a combination of Eqs. 9.234 ad(1) and ad(2) in Ref. 9]

*n*> 0. Set, for

*k*= 0, 1, …,

*n*,

*A*(0) coincides with the LHS of Eq. (35).

*A*(

*k*+ 1) =

*A*(

*k*) for

*k*= 0, 1, …,

*n*− 1. Suppose 0 ≤

*k*<

*n*. Then

*A*(

*k*+ 1) −

*A*(

*k*) = 0.

*A*(

*n*) we apply substitutions

*κ*′ =

*κ*− 1/2,

*μ*′ =

*μ*+ 1/2, and obtain, by the induction hypothesis,

*A*(0) =

*A*(

*n*), the identity follows.□

Here is a generalization of the addition formula for the Whittaker functions (25).

*For*0 ≤

*r*

_{0}<

*r*

*,*$\kappa ,\mu \u2208C$

*,*Re

*μ*> 0

*,*

The equation for *r*_{0} = 0 should be understood as a limiting case of (37), and it is trivial.

*r*

_{0}<

*r*. Let us shortly analyze this point. Put

*ℓ*,

But on the other hand, the series turns out to be numerically quite unstable for large values of *μ*. In such a case we can be dealing with an alternating series with many summands in its beginning attaining huge values. Then significant cancellations of the terms necessarily happen. From the numerical point of view this is a troublesome situation. As an example let us consider the case with *μ* = 20, *κ* = 1, *r*_{0} = 1, and *r* = 2. The Computer Algebra System *Mathematica*, as of version 14.0.0, gives the values *t*_{0} = 1.072 39 × 10^{7}, *t*_{145} = 3214.65, and is not capable to compute *t*_{ℓ} for higher indices. Nonetheless replacing the involved Whittaker functions by their leading asymptotic terms for *μ* large one finds that the first index for which *t*_{ℓ} attains a value smaller than 0.1 is *ℓ* = 168. This shows that this concrete series starts to rapidly converge to its final sum only for very large summation indices.

*r*

_{0}we obtain an equivalent countable system of equations, with $n\u2208Z+$,

*U*in terms of the Whittaker function

*W*we get

Finally we prove a summation formula for the Whittaker functions *M*_{κ,μ}.

*For*$z\u2208C$

*,*$\kappa \u2208C$

*,*

*μ*> 0

*and*

*c*∈ [−1, 1]

*,*

*or, if rewritten in terms of the Whittaker functions, with*

*γ*∈ [−

*π*,

*π*]

*,*

Here we tacitly assume that *z*^{−μ−1/2}*M*_{κ,ℓ+μ}(*z*) is understood as an entire function – first defined on the positive half-line and then admitting an unambiguous continuation to the entire complex plane as an analytic function.

*γ*= 0 gives

*γ*=

*π*gives

^{6}For our purposes a comparatively simple approach is sufficient. From equation (3.30) in Ref. 2 one infers that

*ν*> 0, 0 ≤

*θ*≤

*π*and $\u2113\u2208Z+$. It follows that

*c*∈ [−1, 1] we have

*ℓ*sufficiently large, surely for any

*ℓ*such that

*ℓ*+ 2

*μ*+ 1 > |

*μ*−

*κ*+ 1/2|, we can estimate

*K*(

*κ*,

*μ*) is a constant independent of

*ℓ*and

*z*although it may depend on

*κ*and

*μ*. The convergence now becomes obvious.

These estimates even show that the LHS of (39) is an entire function.

*z*we get an equivalent countable set of equations, with $n\u2208Z+$,

*c*= 1 − 2

*w*we get the equation

*w*leads to the equations

*ℓ*→

*ℓ*+

*j*and applying the substitutions

*n*=

*N*+

*j*,

*ν*=

*μ*+

*j*, we obtain an identity [Eq. (42)] which is proven in Lemma 13 below thus concluding this proof.□

*For*$N\u2208Z+$

*and*$\nu \u2208C$

*,*

*ν*≠ 0, −1, −2, …

*,*

*ν*→

*∞*, equals

*ν*= 0, −1/2, −1, …, − (2

*N*− 1)/2, −

*N*, and all of them are of first order.

*ν*+ 1/2)

_{N}the singularities are removable for

*ν*= 0, −1, …, −

*N*. The residue at a pole

*ν*= −

*t*, 0 ≤

*t*≤

*N*, equals

*t*+ 1/2)

_{N}. Shifting the summation index

*ℓ*→

*t*+

*ℓ*, for

*t*≥

*N*/2 we get

*ℓ*. For

*t*≤

*N*/2 we similarly get

## ACKNOWLEDGMENTS

The author acknowledges partial support by European Regional Development Fund Project “Center for Advanced Applied Science” Grant No. CZ.02.1.01/0.0/0.0/16_019/0000778. The author is indebted to the reviewer for numerous comments helpful in improving the quality of the paper, and in particular for Remark 9.

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### Author Contributions

**Pavel Šťovíček**: Investigation (lead).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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