Given a weighted *ℓ*^{2} space with weights associated with an entire function, we consider pairs of weighted shift operators, whose commutators are diagonal operators, when considered as operators over a general Fock space. We establish a calculus for the algebra of these commutators and apply it to the general case of Gelfond–Leontiev derivatives. This general class of operators includes many known examples, such as classic fractional derivatives and Dunkl operators. This allows us to establish a general framework, which goes beyond the classic Weyl–Heisenberg algebra. Concrete examples for its application are provided.

## I. INTRODUCTION

It is a well-known fact that the Bargmann–Fock space arises via the Bargmann representation of the Heisenberg group. This results in a close connection between these two; see Refs. 8, 15, or 18. Moreover, this fact also explains why the classic Bargmann–Fock space is a central object in quantum physics. The importance of the Bargmann–Fock space stems from the fact that in this space the dual of the derivative operator is the multiplication operator *M*_{z}. Since these operators are Fourier (as well as Fischer) duals of each other, it provides the basis for the study of pseudo-differential operators on Fock spaces and of Toeplitz operators. Furthermore, the Lie group induced by the arising Lie algebra has some interesting consequences, such as its translation invariance.

This automatically leads to the question if we can use the same or similar algebraic methods in other settings where the duals with respect to the integral transform are different from the above. Instead of concentrating on the derivative and multiplication operators whose commutator is the identity and, therefore, immediately leads to the Heisenberg group,^{8} we first look at the dual pair of integration and back-shift operators. However, the commutator between the two is not the identity operator, but rather a (infinite-dimensional) diagonal operator when applied to the standard (non-normalized) basis *e*_{n}(*z*) = *z*^{n}. This is an important point since in many other cases, such as fractional derivatives of Gelfond–Leontiev type, we encounter similar algebraic properties. In fact, as will be seen later, the class of Gelfond–Leontiev operators of generalized differentiation provides a general setting, which includes standard examples, such as classic fractional derivatives (such as Caputo and Riemann–Liouville derivatives, the latter by changing the ground state^{11}), Dunkl (or difference-differential) operators,^{16} and the *q*-derivative,^{5,12,13,17} and, therefore, has a broad range of applications in physics. We only recall that while fractional differential operators are being applied in many areas, such as fractional mechanics, Dunkl operators appear naturally in the study of Calogero–Sutherland–Moser models for *n*-particle systems.^{16} In these cases, even the commutator of the derivative and multiplication operator *M*_{z} does not lead to the identity operator, but to a diagonal operator when applied to the standard basis. The one exception is the case of the *q*-derivative where we can replace the commutator by the *q*-commutator as an alternative. In our general case of fractional derivatives, this leads to the well-known fact that, in general, there is no finite Leibniz rule for fractional derivatives. While this seems to be a problem, it also means that the resulting Lie algebra is much richer. Therefore, we need an approach, which allows us to work with such structures where the commutator is a diagonal operator when applied to an appropriate basis. To this end, we are going to study commutator relations over generalized Fock spaces. As a principal example, we are going to look into the important case of the backward-shift operator, which is dual of the integration operator in the standard case. This allows us to consider settings that *a priori* are quite different, but which fit in the general analysis we are doing.

The outline of this paper is as follows. In Sec. II, we discuss the backward shift operator in the classical Fock space. In Sec. III, we study the Fock space associated with the Gelfond–Leontiev operator. A first example of the calculus on the diagonal is given in Sec. IV, where the family of spaces studied in Ref. 4 is considered. Finally, in Sec. V, we consider the general setting of pairs of weighted shift operators.

## II. THE BACKWARD SHIFT OPERATOR IN THE FOCK SPACE

By starting with the backward-shift operator, we can calculate its dual in Fock spaces with more general weights than usually considered in the literature, but which continue to be reproducing kernel Hilbert spaces.

*∂*and the operator

*M*

_{z}of multiplication by the variable

*z*are closed operators defined on the span of polynomials and adjoint to each other there. Their commutator

**id**. In fact, it then follows that the elements are entire functions; see Refs. 6 and 7.

*R*

_{0}defined by

*R*

_{0}is a contraction from the Fock space into itself, with its adjoint being the integration operator

*I*defined by

*For the case of the classic Fock space* $F1$, *we have for the backward-shift operator* *R*_{0} *the following properties:*

*R*_{0}*is a contraction in the Fock space.**Its adjoint operator*$R0*$*is the integration operator.*

*f*is an entire function $f(z)=\u2211n=0\u221efnzn$ with a finite $F1$-norm,

*R*

_{0}is bounded, it is enough to check that $R0*=I$ on monomials. Let $n,m\u2208N0$. We have

*m*≥ 1,

*m*≥ 1. For

*m*= 0, the equality is trivial,

*R*

_{0}and

*I*is equal on monomials to

*D*

_{0}the (formal) diagonal operator defined by the right-hand side of (2.4), i.e.,

*R*

_{0}and

*I*also requires diagonal operators as coefficients.

## III. FOCK SPACES RELATED TO FRACTIONAL DERIVATIVES

*ρ*> 0 and degree

*σ*> 0, that is, such that $limn\u2192\u221en1\rho |\phi n|n=\sigma e\rho 1\rho $.

*Assume that*

*φ*

_{n}> 0

*for*$n\u2208N0$

*. We define the Hilbert space*$H(\phi )$

*as the set of all entire functions endowed with the inner product,*

*where*$f(z)=\u2211n=0\u221efnzn$

*and*$g(z)=\u2211n=0\u221egnzn$

*.*

*The Hilbert space*$H(\phi )$

*is a reproducing kernel Hilbert space with the reproducing kernel*

When the sequence *φ*_{n} is increasing and *φ*_{0} = 1, the space $H(\phi )$ is a de Branges–Rovnyak space. See Ref. 1.

*Let*

*φ*

*be as in*

*(3.1)*

*. We define the Gelfond–Leontiev (G–L) operator of generalized differentiation with respect to*

*φ*

*, denoted as*

*∂*

^{φ}

*, as the operator acting on function*

*f*

*analytic in a neighborhood of the origin,*$f(z)=\u2211n=0\u221efnzn$

*, by*

It is well known that $\u2202\phi *=Mz$ over the space $H(\phi )$. We now introduce the corresponding generalization of the integration operator *I*.

*The operator*

*R*

_{0}

*is densely defined and closed in*$H(\phi )$

*. It is a contraction when the sequence*$(\phi n)n=0\u221e$

*is non-decreasing. The adjoint of*

*R*

_{0}

*is given by*

*R*

_{0}is closed due to the fact that in a reproducing kernel Hilbert space, convergence in norm implies pointwise convergence. We consider a sequence of functions $f\u0303k:C\u21a6C$ in $H(\phi )$ converging in norm to $f\u2208H(\phi )$ and such that the sequence $(R0f\u0303k)k=1\u221e$ is convergent, with limit $g\u2208H(\phi )$. Then, for every

*ω*≠ 0 in the domain of analyticity of the elements of $H(\phi )$,

*g*(

*ω*) =

*R*

_{0}

*f*(

*ω*) for

*ω*≠ 0 and for

*ω*= 0 by analytic continuation.

*φ*

_{n}≤

*φ*

_{n+1}for $n\u2208N0$, we have

*R*

_{0}. Let $n,m\u2208N0$. We have

We note that inequality (3.8) is the structure inequality, which characterizes de Branges–Rovnyak spaces; see Ref. 2 (Theorem 3.1.2, p. 83) and Ref. 1 for further on this point.

*Let*

*φ*

*be as in*

*(3.1)*

*. We define the Gelfond–Leontiev (G–L) operator of generalized integration with respect to*

*φ*

*, denoted as*

*I*

^{φ}

*, as the operator acting on functions*

*f*

*analytic in a neighborhood of the origin,*

*φ*(

*z*) =

*e*

^{z}, we have

*φ*

_{n}= Γ(

*n*+ 1),

*n*= 0, 1, 2, …, so that

*φ*is the Mittag–Leffler function,

*ρ*and type 1 [see Ref. 9 (p. 56)]. We obtain $\phi n=\Gamma \mu +n\rho $, and operator (3.6) becomes the Dzrbashjan–Gelfond–Leontiev operator,

*The function* $E1\rho ,\mu (z\omega \u0304)$ *is positive definite if* *μ* > 0*.*

This is an immediate consequence since for *ρ*, *μ* > 0, we have $\Gamma (\mu +n\rho )>0$ and, therefore, *φ*_{n} > 0 too.

*σ*(

*x*) = −

*x*. Given a multiplicity constant $\kappa \u2208C(Re(\kappa )>0)$, we get the first-order rational Dunkl operator attached to $G$ and

*κ*defined as

*φ*with

*T*can be seen as a special case of the Gelfond–Leontiev operator

*∂*

^{φ}. Moreover, the function $\phi (z\omega \u0304)$ is positive definite since coefficients (3.13) are positive. Likewise, we obtain

*q*∈ (0, 1). The case where

*φ*

_{n}is the

*q*deformation of

*n*!, that is,

*j*> 0 and $0q=1$, corresponds to the case of the so-called

*q*-calculus. We refer, in particular, to the papers

^{5,12,13,17}for a study of the corresponding Fock spaces and commutations relations.

## IV. DIAGONAL OPERATORS OVER THE SPACES $Fp$

*p*-Fock space. These spaces have been investigated.

^{4}For us, they are particularly interesting since they correspond to the choice of

*φ*

_{n}= (

*n*!)

^{p}for $p\u2208N$ or, in other words, they represent the case of powers of the factorials.

*We define the*

*p*

*-Fock space*$Fp,p\u2208N$

*, as the set of all entire functions*$f:C\u21a6C$

*such that*

*p*= 1 is the classic Fock space $F1$ already mentioned in Sec. I. The inner product between two functions $f,g\u2208Fp$ is given by

*p*-Fock space $Fp$ is a reproducing kernel Hilbert space with reproducing kernel

Note that the space $F2$ was considered in Ref. 10 (Lemma 4, p. 181) and plays an important role in the theory of discrete analytic functions; see Ref. 3 for details.

In these spaces, the backward-shift operator has the following properties.

*Let*$p\u2208N$

*. Then, we have the following:*

*The backward-shift operator**R*_{0}*is a contraction in*$Fp$*.**Its adjoint operator*$R0*$*is given by*

*where*

*I*

*stands for the integration operator*

*(2.3)*

*.*

The proof that *R*_{0} is bounded in $Fp,p=2,3,\u2026$, follows the same lines as the proof in the classic case *p* = 1.

In Ref. 4, it was shown that for the multiplication operator and the derivative operator, i.e., with *A* = *M*_{z} and *B* = *∂*_{z}, we have a similar formula for the adjoint *A** = (*BA*)^{p−1}*B* in $Fp$.

Now, we want to discuss the commutation relations in terms of diagonal operators.

*D*

^{(1)}

*z*

^{n}=

*d*

_{n−1}

*z*

^{n}(under the convention that

*d*

_{−1}= 0) and

*D*

^{(−1)}

*z*

^{n}=

*d*

_{n+1}

*z*

^{n}. We note that

*P*≔ diag(0, 1, 1, 1, …).

*R*

_{0}nor

*I*belongs to $D$, we have $[R0,I]=D0\u2208D$, where

*D*

_{0}is the diagonal operator (2.5) linked to the Fock space $F1$.

This allows us to state the following lemma.

*On the linear span of the polynomials, it holds that*

*DI*=*ID*^{(−1)}*,**ID*=*D*^{(1)}*I**,**DR*_{0}=*R*_{0}*D*^{(1)}*, and**D*^{(−1)}*R*_{0}=*R*_{0}*D**for every*$D\u2208D$*.*

Under the usual convention 0*z*^{−1} = 0, we have the following:

$DIzn=Dzn+1n+1=dn+1zn+1n+1=I(dn+1zn)=ID(\u22121)zn$,

$IDzn=I(dnzn)=dnzn+1n+1=D(1)(zn+1n+1)=D(1)Izn$,

*DR*_{0}*z*^{n}=*Dz*^{n−1}=*d*_{n−1}*z*^{n−1}=*R*_{0}(*d*_{n−1}*z*^{n}) =*R*_{0}*D*^{(1)}*z*^{n}, and*D*^{(−1)}*R*_{0}*z*^{n}=*D*^{(−1)}*z*^{n−1}=*d*_{n}*z*^{n−1}=*R*_{0}*d*_{n}*z*^{n}=*R*_{0}*Dz*^{n}for $n\u2208N0$.

*m*,

*m*-shift operators, as well as

*R*

_{0}

*I*,

*IR*

_{0}, are diagonal operators, they commute. Hence, in the same way as in the previous lemma, we have

*M*

_{α,q}defined in Ref. 5 (Theorem 4.3).

However, we need an auxiliary lemma before stating our main results.

*For every*$D\u2208D$

*and*$n,k\u2208N0$,

*it holds that*

While we state the lemma in a general form of principal importance, for us will be the case of *n* = 1.

*For*

*R*

_{0}

*and*

*I*

*acting on the linear span of polynomials and*

*D*

_{0}

*defined by*

*(2.5)*

*, it holds that*

*where*$\Lambda k,n\u2208D$

*are given by the recurrence relation,*

*with initial values*

*and with*

*for remaining values*

*k*,

*n*

*.*

*n*= 1, we have

_{1,1}= diag(1, 1, 1, …).

*n*,

_{k,n+1}are defined by formula (4.11) at rank

*n*for

*j*,

*k*= 1, …,

*n*and Λ

_{n+1,n+1}is the identity diagonal. Therefore, $(IR0)n+1=\u2211k=1n+1\Lambda k,n+1IkR0k$, and the result holds for all integers.□

Note that the same follows for $(R0I)n$, albeit with $D0(l)$ replaced by $D0(\u2212l)$. To make it easier to get a clear idea, we now illustrate (4.10) and (4.11) for the cases *n* = 2 and *n* = 3.

**Case***n*= 3**:**Iterating (2.6) and using the case*n*= 2, we now haveso that$(IR0)3=(IR0)2(IR0)=(I2R02+D0(1)IR0)(IR0)=I2R0(R0I)R0+D0(1)(I2R02+D0(1)IR0)=I2R0(IR0+D(1))R0+D0(1)I02R02+(D0(1))2IR0=I2(IR0+D0)R02+D0(1)I2R02+D0(1)(I02R02+D0(1)IR0)=I3R03+(D0(2)+2D0(1))I2R02+D0(1)IR0$Equation (4.11) for(4.14)$\Lambda 3,3=id,\Lambda 2,3=2D0(1)+D0(2),and\Lambda 1,3=(D0(1))2.$*n*= 3 and*k*= 1 and*k*= 2, respectively, becomeswhich are verified by (4.14) in view of (4.13)-(4.13).$\Lambda 1,3=\Lambda 0,2+D0(1)\Lambda 1,2,\Lambda 2,3=\Lambda 1,2+(D0(1)+D0(2))\Lambda 2,2,$

We now compute the coefficients Λ_{k,n} in terms of *D*_{0}.

*For*$D0(l)$

*as defined above,*

*where*|

*α*| ≔

*α*

_{1}+ ⋯ +

*α*

_{k}

*and*

*α*

_{i}

*are non-negative integers.*

_{k,n}. If

*k*= 0, there are no elements to sum, and thus, Λ

_{0,n}= 0. Now, for

*n*≥ 1,

*α*

_{i}gives the following:

*α*

_{k+1}= 0 for $\u2211t=1k+1\alpha t=n\u2212k$, while the second sum can be seen as all terms in which

*α*

_{k+1}> 0 for $\u2211t=1k+1\alpha t=n\u2212k$. Thus, the equality proposed above holds and this is a valid representation for the exact form of Λ

_{k,n}.□

*For*0 <

*k*<

*n*,

*we have the formula for the entries of the matrices*Λ

_{k,n},

*i*> 2, we obtain the following:

- If
*k*≥*i*− 1, then$\u220ft=1k\u2211l=1tD0(l)\alpha ti,i=\u220ft=i\u22121k\u2211l=1tD0(l)\alpha t\u220ft=1i\u22122\u2211l=1tD0(l)\alpha ti,i=1i\u22121\alpha i\u22121+\cdots +\alpha k(\u22121)\alpha 1+\cdots +\alpha i\u221221(i\u22121)\alpha 1+\cdots +\alpha i\u221221\alpha 12\alpha 2\cdots (i\u22122)\alpha i\u22122(i\u22122)\alpha 1(i\u22123)\alpha 2\cdots 1\alpha i\u22122=1i\u22121n\u2212k\u220fs=1i\u22122\u2212si\u22121\u2212s\alpha s.$ - If
*k*<*i*− 1, then$\u220ft=1k\u2211l=1tD0(l)\alpha ti,i=(\u22121)\alpha 1+\cdots +\alpha k1(i\u22121)\alpha 1+\cdots +\alpha k1\alpha 12\alpha 2\cdots k\alpha kk\alpha 1(k\u22121)\alpha 2\cdots 1\alpha k=1i\u22121n\u2212k\u220fs=1k\u2212si\u22121\u2212s\alpha s.$

For future computations, it will be convenient to introduce the shifted version of Λ_{k,n} under the map (4.4).

*Let*$\Gamma k,n=\Lambda k,n(\u22121)$

*. Then,*

*Then,*

*and*

*with the same boundary conditions*

*(4.12)*

*and*

*(4.13)*

*as for*Λ

_{k,n}

*.*

*In terms of the entries of the matrices*Γ

_{k,n}(0 <

*k*<

*n*),

*we have the formula*

*i*≥ 2 we obtain the following:

- If
*k*≥*i*, then$\u220ft=1k\u2211l=0t\u22121D0(l)\alpha ti,i=\u220ft=ik\u2211l=0t\u22121D0(l)\alpha t\u220ft=1i\u22121\u2211l=0t\u22121D0(l)\alpha ti,i=1i\alpha i+\cdots +\alpha k(\u22121)\alpha 1+\cdots +\alpha i\u22121i\alpha 1+\cdots +\alpha i\u221211\alpha 12\alpha 2\cdots (i\u22121)\alpha i\u22121(i\u22121)\alpha 1(i\u22122)\alpha 2\cdots 1\alpha i\u22121=1in\u2212k\u220fs=1i\u22121\u2212si\u2212s\alpha s.$ - If
*k*<*i*, then$\u220ft=1k\u2211l=0t\u22121D0(l)\alpha ti,i=1i\alpha 1+\cdots +\alpha k(\u22121)\alpha 1(\u22122)\alpha 2\cdots (\u2212k)\alpha k(i\u22121)\alpha 1(i\u22122)\alpha 2\cdots (i\u2212k)\alpha k=1in\u2212k\u220fs=1k\u2212si\u2212s\alpha s.$

These considerations allow us to compute the commutator of *R*_{0} and its adjoint. Recall that *D*_{0} defined by (2.5) satisfies [*R*_{0}, *I*] = *D*_{0}. In $F1$, we have *I** = *R*_{0} and so $[R0*,R0]=\u2212D0$ in that space. We now compute this commutator in a general $Fp$ space.

*For any*$Fp$

*as defined above,*

_{p,p−1}= 0. Now, we compute

## V. THE GENERAL CASE

*A*and

*B*on $H(\phi )$,

*The conclusions of Lemma 4.4 are still valid with* *A* *instead of* *R*_{0} *and* *B* *instead of* *I**, namely, we have the following:*

*DB*=*BD*^{(−1)},*BD*=*D*^{(1)}*B*,*DA*=*AD*^{(1)},*and**AD*=*D*^{(−1)}*A*.

*n*≥ 1, we have

*D*(

*a*,

*b*). We have

*A*and

*B*are densely defined and closed.

*It holds that*

*where*Λ

_{k,n}

*is defined as before, with*

*D*=

*D*(

*a*,

*b*) = [

*A*,

*B*]

*.*

□

*The operators*

*A*

*and*

*B*

*satisfy*

*A** =

*B*

*if and only if*

*m*≥ 1,

*n*=

*m*+ 1, we have

An immediate consequence of the above theorem is the following.

*The set of sequences* (*a*, *b*) *satisfying* *(5.5)* *is a real infinite-dimensional vector space.*

We have the following:

For

*A*=*∂*and*B*=*M*_{z}, we have*a*_{n}=*n*and*b*_{n}= 1 and (5.5) becomes (*n*+ 1)*φ*_{n}=*φ*_{n+1}, that is,*φ*_{n}=*c*·*n*! for some*c*> 0, and we get the Fock space, as expected.For

*A*=*R*_{0}and*B*=*I*, we have*a*_{n}= 1 and $bn=1n+1$ and (5.5) becomes now $\phi n=\phi n+1n+1$, which also leads to the same*φ*_{n}as above.If

*A*=*R*_{0}and*B*=*M*_{z}, we have*a*_{n}= 1 and*b*_{n}= 1 and*φ*_{n}=*c*. Here, we get the Hardy space, but we are not in the setting of entire functions anymore.

Since we assumed the operators to be closed, *A** = *B* is equivalent to *A* = *B**.

*Let*$p\u2208N$

*,*

*p*≥ 2

*. Then, the operators*

*A*

*and*

*B*

*defined satisfy*

*A** = (

*BA*)

^{p−1}

*B*

*if and only if*

*A**

*z*

^{n},

*z*

^{m}⟩ in the Proof of Theorem 5.3. On the other hand, for

*n*≥ 1,

*BAz*

^{n}=

*B*(

*a*

_{n}

*z*

^{n−1}) =

*a*

_{n}

*b*

_{n−1}

*z*

^{n}. Thus,

*m*=

*n*+ 1,

*Let*$p\u2208N$

*,*

*p*≥ 2

*. Then, the operators*

*A*

*and*

*B*

*satisfy*

*B** = (

*AB*)

^{p−1}

*A*

*if and only if*

*ABz*

^{n}=

*A*(

*b*

_{n}

*z*

^{n+1}) =

*a*

_{n+1}

*b*

_{n}

*z*

^{n}for $n\u2208N0$. Thus, for

*n*≥ 1,

*n*=

*m*+ 1, we obtain

Thus, given such an operator *A*, these formulas can be used to find an associated integration operator such that the adjoint of A can be expanded in this way. Similarly, given B, one can find an associated differentiation operator for this expansion.

*A*and

*B*satisfying

*A** = (

*BA*)

^{m−1}

*B*and with

*D*= [

*A*,

*B*], we have

*B** = (

*AB*)

^{m−1}

*A*and with

*D*= [

*B*,

*A*], we have

*D*is the identity. In this case, Λ

_{k,n}= Γ

_{k,n}=

*S*(

*n*,

*k*), and this expansion simplifies to

*B*=

*M*

_{z}and

*A*=

*∂*

_{z}. The recurrence relation for these numbers can then be reformatted to yield

*D*to the identity operator, then Λ

_{k,n}= Γ

_{k,n}=

*S*(

*n*,

*k*), where

*S*(

*n*,

*k*) are the Stirling numbers of the second kind. The recurrence relation for these numbers can then be rewritten to yield

*A*=

*R*

_{0}and

*B*=

*I*

^{φ}, we have

*D*= [

*R*

_{0},

*I*

^{φ}], which is a diagonal operator,

*φ*

_{−1}= 0.

This leads to the following expressions of the matrix entries for Λ_{k,n} and Γ_{k,n}.

*In terms of the entries of both matrices*Λ

_{k,n}

*and*Γ

_{k,n}(0 <

*k*<

*n*),

*we have*

*and*

_{k,n}-entries the following:

- If
*i*= 2, then$\u220ft=1k\u2211l=1tD(l)\alpha ti,i=\phi 0\phi 1n\u2212k.$ - If 2 <
*i*≤*k*+ 1, then$\u220ft=1k\u2211l=1tD(l)\alpha ti,i=\u220ft=i\u22121k\u2211l=1tD(l)\alpha t\u220ft=1i\u22122\u2211l=1tD(l)\alpha ti,i=\phi i\u22122\phi i\u22121\alpha i\u22121+\cdots +\alpha k\u220ft=1i\u22122\phi i\u22122\phi i\u22121\u2212\phi i\u2212t\u22122\phi i\u2212t\u22121\alpha t.$ - If
*i*>*k*+ 1, thenTherefore, the result holds true for the Λ$\u220ft=1k\u2211l=1tD(l)\alpha ti,i=\u220ft=1k\phi i\u22122\phi i\u22121\u2212\phi i\u2212t\u22122\phi i\u2212t\u22121\alpha t.$_{k,n}-entries. In a similar way, we have for the Γ_{k,n}-entries thatso that the following holds:$\u2211l=0t\u22121D(l)i,i=\phi i\u22121\phi i\u2009if\u20091\u2264i\u2264t,\phi i\u22121\phi i\u2212\phi i\u2212t\u22121\phi i\u2212t\u2009if\u2009i>t$

- If
*i*= 1, then$\u220ft=1k\u2211l=0t\u22121D(l)\alpha ti,i=\phi 0\phi 1n\u2212k.$ - If 1 <
*i*≤*k*, then$\u220ft=1k\u2211l=0t\u22121D(l)\alpha ti,i=\u220ft=ik\u2211l=0t\u22121D(l)\alpha t\u220ft=1i\u22121\u2211l=0t\u22121D(l)\alpha ti,i=\phi i\u22121\phi i\alpha i+\cdots +\alpha k\u220ft=1i\u22121\phi i\u22121\phi i\u2212\phi i\u2212t\u22121\phi i\u2212t\alpha t.$ - If
*i*>*k*, then$\u220ft=1k\u2211l=0t\u22121D(l)\alpha ti,i=\u220ft=1k\phi i\u22121\phi i\u2212\phi i\u2212t\u22121\phi i\u2212t\alpha t.$

We conclude this section with an application of Proposition 5.10.

_{k,n}the following:

$(\Lambda k,n)1,1=0$.

$(\Lambda k,n)2,2=n\u2212k+1k\u2212111+2\kappa n\u2212k$.

- If 2 <
*i*≤*k*+ 1,*i*being even, we have$(\Lambda k,n)i,i=\u2211|\alpha |=n\u2212k\phi i\u22122\phi i\u22121\alpha i\u22121+\cdots +\alpha k\u220ft=1i\u22122\phi i\u22122\phi i\u22121\u2212\phi i\u2212t\u22122\phi i\u2212t\u22121\alpha t=\u2211|\alpha |=n\u2212k1i\u22121+2\kappa \alpha i\u22121+\cdots +\alpha k\u220fl=1i/2\u221211i\u22121+2\kappa \u22121i\u22122l\alpha 2l\u22121\xd7\u220fl=1i/2\u221211i\u22121+2\kappa \u22121i\u22121\u22122l+2\kappa \alpha 2l=1i\u22121+2\kappa \alpha n\u2212k\u2211|\alpha |=n\u2212k\u220fl=1i/2\u221211\u22122l\u22122\kappa i\u22122l\alpha 2l\u22121\u22122li\u22121\u22122l+2\kappa \alpha 2l,$*i*being odd, we have

$(\Lambda k,n)i,i=\u2211|\alpha |=n\u2212k\phi i\u22122\phi i\u22121\alpha i\u22121+\cdots +\alpha k\u220ft=1i\u22122\phi i\u22122\phi i\u22121\u2212\phi i\u2212t\u22122\phi i\u2212t\u22121\alpha t=\u2211|\alpha |=n\u2212k1i\u22121\alpha i\u22121+\cdots +\alpha k\u220fl=1(i\u22121)/21i\u22121\u22121i\u22122l+2\kappa \alpha 2l\u22121\xd7\u220fl=1(i\u22121)/2\u221211i\u22121\u22121i\u22121\u22122l\alpha 2l=1i\u22121\alpha n\u2212k\u2211|\alpha |=n\u2212k2\u2212i+2\kappa 1+2\kappa \alpha i\u22122\u220fl=1(i\u22121)/2\u221211\u22122l+2\kappa i\u22122l+2\kappa \alpha 2l\u22121\u22122li\u22121\u22122l\alpha 2l.$ - If
*i*>*k*+ 1,and*i*being even, we have for*k*= 2*m*,

while for$(\Lambda k,n)i,i=\u2211|\alpha |=n\u2212k\u220ft=1k\phi i\u22122\phi i\u22121\u2212\phi i\u2212t\u22122\phi i\u2212t\u22121\alpha t=\u2211|\alpha |=n\u22122m\u220fl=1m1i\u22121+2\kappa \u22121i\u22121\u22122l+2\kappa \alpha 2l1i\u22121+2\kappa \u22121i\u22122l\alpha 2l\u22121=\u22121i\u22121+2\kappa n\u22122m\u2211|\alpha |=n\u22122m\u220fl=1m2li\u22121\u22122l+2\kappa \alpha 2l2l+2\kappa \u22121i\u22122l\alpha 2l\u22121,$*k*= 2*m*+ 1, we get$(\Lambda k,n)i,i=\u2211|\alpha |=n\u2212k\u220ft=1k\phi i\u22122\phi i\u22121\u2212\phi i\u2212t\u22122\phi i\u2212t\u22121\alpha t=\u2211|\alpha |=n\u22122m\u22121\u220fl=1m1i\u22121+2\kappa \u22121i\u22121\u22122l+2\kappa \alpha 2l\u220fl=1m+11i\u22121+2\kappa \u22121i\u22122l\alpha 2l\u22121=\u22121i\u22121+2\kappa n\u22122m\u22121\u2211|\alpha |=n\u22122m\u221212m+2\kappa +1i\u22122m\u22122\alpha 2m+1\u220fl=1m2li\u22121\u22122l+2\kappa \alpha 2l2l+2\kappa \u22121i\u22122l\alpha 2l\u22121,$*i*being odd, we have for*k*= 2*m*,

while for$(\Lambda k,n)i,i=\u2211|\alpha |=n\u2212k\u220ft=1k\phi i\u22122\phi i\u22121\u2212\phi i\u2212t\u22122\phi i\u2212t\u22121\alpha t=\u2211|\alpha |=n\u22122m\u220fl=1m1i\u22121\u22121i\u22121\u22122l\alpha 2l1i\u22121\u22121i\u22122l+2\kappa \alpha 2l\u22121=\u22121i\u22121n\u22122m\u2211|\alpha |=n\u22122m\u220fl=1m2li\u22121\u22122l\alpha 2l2l\u22122\kappa \u22121i\u22122l+2\kappa \alpha 2l\u22121,$*k*= 2*m*+ 1, we get$(\Lambda k,n)i,i=\u2211|\alpha |=n\u2212k\u220ft=1k\phi i\u22122\phi i\u22121\u2212\phi i\u2212t\u22122\phi i\u2212t\u22121\alpha t=\u2211|\alpha |=n\u22122m\u22121\u220fl=1m1i\u22121\u22121i\u22121\u22122l\alpha 2l\u220fl=1m+11i\u22121\u22121i\u22122l+2\kappa \alpha 2l\u22121=\u22121i\u22121+2\kappa n\u22122m\u22121\u2211|\alpha |=n\u22122m\u221212m\u22122\kappa +3i\u22122m\u22122+2\kappa \alpha 2m+1\u220fl=1m2li\u22121\u22122l\alpha 2l2l\u22122\kappa \u22121i\u22122l+2\kappa \alpha 2l\u22121.$

## VI. CONCLUSIONS

In this paper, we provided a general framework on how to handle commutators of operators, which act differently according to the basis elements. Using our approach of working with diagonal operators, we could obtain representations of the commuting relation both as operators and action on the basis elements. This allows for a variety of avenues to expand upon these results. For our concrete formulas, the considered operators are those that shift the basis element by one power, such as multiplication and backward shift operators. Our method also allows us to consider more general shift operators.

In terms of applications, the obtained framework can be used not only to calculate the corresponding Lie algebras but also to extend methods and results from the case of the standard Fock space to more general situations involving Gelfond–Leontiev derivatives as considered in the paper. In particular, questions such as pseudodifferential operators or function theories based on Fischer duality can be considered.

## ACKNOWLEDGMENTS

Daniel Alpay acknowledges the Foster G. and Mary McGaw Professorship in Mathematical Sciences, which supported this research. He also thanks Professor Alain Yger for introducing him to the work of Ore.^{14}

P. Cerejeiras and U. Kähler were supported by Portuguese funds through the CIDMA—Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT–Fundação para a Ciência e a Tecnologia”) within Project Nos. UIDB/04106/2020 and UIDP/04106/2020.

Trevor Kling acknowledges the Schmid College of Science and Technology (Chapman University) for an undergraduate summer research grant.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Daniel Alpay**: Conceptualization (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). **Paula Cerejeiras**: Conceptualization (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). **Uwe Kähler**: Conceptualization (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). **Trevor Kling**: Conceptualization (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal).

## 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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