On the role of Hermite-like polynomials in the Fock representations of Gaussian states

TL;DR

This paper explores how Hermite-like polynomials can be used to represent Gaussian states and operators in quantum mechanics, providing explicit formulas that simplify calculations in continuous-variable quantum systems.

Contribution

It introduces a novel approach using Hermite-like polynomials for Fock representations of Gaussian states, yielding closed-form algebraic results.

Findings

Explicit formulas for Gaussian states in Fock basis

Simplified evaluation of quantum states and operators

Fundamental role of Hermite-like polynomials in quantum representations

Abstract

The expansion of quantum states and operators in terms of Fock states plays a fundamental role in the field of continuous-variable quantum mechanics. In particular, for general single-mode Gaussian operators and Gaussian noisy states, many different approaches have been used in the evaluation of their Fock representation. In this paper a natural approach has been applied using exclusively the operational properties of the Hermite and Hermite-like polynomials and showing their fundamental role in this field. Closed-form results in terms of polynomials, exponentials, and simple algebraic functions are the major contribution of the paper.

Equations

$$H_n(x,y)$$

$$e^{y{\partial }_x}g(x)=\sum _{n=0}^{\infty }\frac{y^n}{n!}{\partial }_x^{n}g(x)=g(x+y)$$

$$H_n(x,y)=e^{y{\partial }_x^2}{x}^n.$$

$$e^{y{\partial }_x^2}{x}^n=\sum _{r=0}^{\infty }\frac{y^r}{r!}{\partial }_x^{2r}{x}^n$$

$${\partial }_x^{2r}{x}^n=\frac{n!}{(n-2r)!}{x}^{n-2r}$$

$$e^{y{\partial }_x^2}{x}^n=n!\sum _{r=0}^{\lfloor n/2\rfloor }\frac{1}{(n-2r)!r!}{x}^{n-2r}{y}^r=H_n(x,y).$$

$$h(x,y,t)$$

$$=\sum _{n=0}^{\infty }\frac{y^n}{n!}{\partial }_x^{2n}{e}^{tx}=e^{tx}\sum _{n=0}^{\infty }\frac{y^n{t}^{2n}}{n!}=e^{tx+t^{2}y}.$$

$${\partial }_x{H}_n(x,y)=n{H}_{n-1}(x,y).$$

$$G(x,y;z,v|t)=\sum _{n=0}^{\infty }\frac{t^n}{n!}{H}_n(x,y)H_n(z,v)=\frac{1}{\sqrt{1-4y{t}^{2}v}}\exp \left\{\frac{xtz+t^2(x^{2}v+y{z}^2)}{1-4y{t}^{2}v}\right\}.$$

$$G(x,y;z,v|t)=e^{y{\partial }_x^2+v{\partial }_z^2}\sum _{n=0}^{\infty }\frac{t^n{x}^n{z}^n}{n!}=e^{y{\partial }_x^2+v{\partial }_z^2}{e}^{txz},$$

$$e^{y{\partial }_x^2}f(x)=\frac{1}{2\sqrt{\pi y}}{\int }_{-\infty }^{\infty }{e}^{-\frac{(x-\xi {)}^2}{4y}}f(\xi )d\xi ,$$

$$H_{m,n}(x,y;z,u|\tau )=m!n!\sum _{r=0}^{\min \{m,n\}}\frac{H_{m-r}(x,y)H_{n-r}(z,u){\tau }^r}{(m-r)!r!(n-r)!}.$$

$$H_{m,n}(x,y,z,u|\tau )=e^{y{\partial }_x^2+u{\partial }_z^2+\tau {\partial }_x{\partial }_z}(x^m{z}^n)$$

$$e^{y{\partial }_x^2+u{\partial }_z^2+\tau {\partial }_x{\partial }_z}(x^m{z}^n)$$

$$=e^{\tau {\partial }_x{\partial }_z}{H}_m(x,y)H_n(z,u)$$

$$=\sum _{r=0}^{\infty }\frac{{\tau }^r}{r!}{\partial }_x^r{H}_m(x,y){\partial }_z^r{H}_n(z,u)$$

$$=m!n!\sum _{r=0}^{\min \{m,n\}}\frac{H_{m-r}(x,y){\tau }^r{H}_{n-r}(z,u)}{(m-r)!r!(n-r)!},$$

$$\sum _{m,n=0}^{\infty }\frac{v^m{w}^n}{m!n!}{H}_{m,n}(x,y,z,u|\tau )$$

$$=e^{y{\partial }_x^2+u{\partial }_z^2+\tau {\partial }_x{\partial }_z+vx+wz}.$$

$${}_{2,ϵ}{h}_{m,n}(x,y|\tau )=(m+ϵ)!(n+ϵ)!\sum _{r=0}^{\min \{m,n\}}\frac{x^{m-r}{y}^{n-r}{\tau }^{2r+2ϵ}}{(m-r)!(n-r)!(2r+ϵ)!}$$

$${}_{2,0}{h}_{m,n}(x,y|\tau )$$

$${}_{2,1}{h}_{m,n}(x,y|\tau )$$

$$D(\alpha )=e^{\alpha a^{†}-{\alpha }^{\ast }a},\alpha \in \mathbb{C},$$

$$R(\varphi )=e^{i\varphi a^{†}a},\varphi \in \mathbb{R},$$

$$S(z)=e^{\frac{1}{2}(z{a}^{†}{a}^{†}-z^{\ast }aa)},z=r{e}^{i\theta }\in \mathbb{C},r\ge 0,$$

$$U=K_{0}B(a^{†})C(a^{†},a)F(a)$$

$$K_0=S^{1/2}\exp \left\{-\frac{1}{2}(|\alpha {|}^2+T^{\ast }{\alpha }^2)\right\},B(a^{†})=\exp \left\{\alpha S{a}^{†}+\frac{1}{2}T{a}^{†}{a}^{†}\right\},$$

$$C(a^{†},a)=\sum _{n=0}^{\infty }\frac{(S{e}^{i\varphi }-1{)}^n(a^{†}{)}^n{a}^n}{n!},F(a)=\exp \left\{-(\alpha T^{\ast }+{\alpha }^{\ast })e^{i\varphi }a-\frac{1}{2}{T}^{\ast }{e}^{i2\varphi }{a}^2\right\}.$$

$$U_{m,n}=\frac{K_0}{\sqrt{m!n!}}{H}_{m,n}(x,y;z,u|X)$$

$$x=\alpha S,y=\frac{1}{2}T,z=-(\alpha T^{\ast }+{\alpha }^{\ast })e^{i\varphi },u=-\frac{1}{2}{T}^{\ast }{e}^{i2\varphi },X=S{e}^{i\varphi }$$

$$H_n(x,0)=x^n$$

$$\displaystyle H_{n}(0,y)=\left\{\begin{array}[]{ll}\displaystyle{\frac{y^{n/2}n!}{(n/2)!}}&\mbox{\quad for $n$ even}\\ 0&\mbox{\quad for $n$ odd}\end{array}\right.$$

$$\alpha =0,x=y=z=u=0,X=e^{i\varphi },K_0=1$$

Full text

I Introduction

The representation of quantum operators and states in the infinite dimensional Hilbert space equipped with the Fock-state basisFock32 (called also number-state basis) is a remarkably versatile point of view in the study of continuous-variable quantum mechanics. Even though its application to non Gaussian cases is well documented,Dodo02 it was just in the field of the Gaussian states and operators that Fock representation obtained the most interesting results. The Fock coefficients of the Gaussian density operator $\rho$ have received frequent attention, because its diagonal entries $\rho_{nn}$ give the probability distribution of the number of the photons present in the state.

The mathematics involved in the computations is rather sophisticated and mostly uses the technicalities of the Glauber representation in terms of coherent states (characteristic function, Wigner function, $R$ function, $P$ representation, ecc.).Glau63 ; Cahi69 This implies an inevitable recourse to complicate integrations and/or to frequent quoting of tables of integrals, so that the results are either expressed in terms of hypergeometric functions, Lach65 ; Mari92 or in terms of Laguerre polynomials,Moll67 ; Hels76 or in terms of generalized Hermite polynomials. Vour86 ; Vour87 ; Kim89

In this paper we give a complete and self-contained Fock representation of a general noisy Gaussian single-mode state, based on the properties of the Hermite and Hermite-like (HL) polynomials. Our treatment underlies the interpretation of a Gaussian noisy state as generated by applying a Gaussian unitary transformation to a thermal (often called also chaotic) state. A preliminary computation easily shows that a Gaussian unitary transformation has a very simple Fock representation as a five-variables two-indexes Hermite polynomial, with variables expressed in terms of the bosonic parameters characterizing the displacement, the rotation, and the squeezing implied in the transformation. The passage from the Fock representation of the Gaussian unitary to that of the Gaussian states is conceptually easy, but it requires to sum an infinite series, which may appear a serious drawback (in alternative approaches the drawback is the evaluation of very complicated integrals). But the theory of the HL polynomials through the use of the operational calculus yields the appropriate tool to get a simple closed-form result.

A key point in our derivation is the fact that exponential factors in the normal ordering of the Gaussian unitary have the same structure as the generating function of the two-indices two-variables Hermite polynomials, so that the recourse to the HL polynomial appears natural. As it may be seen in a paperCari15 by two of the authors of the present paper, this holds true also for the multi-mode Gaussian unitaries, provided that the number of indexes and variables in the HL polynomials is appropriately increased.

The paper is organized as follows. In Section II we introduce the different HL polynomials we will use in the following along with their most relevant properties. Their definitions and properties are disseminated in different sources. Then, in order to guarantee the self-consistency of the paper and to clarify the underlying mathematical methodology, we accompany the properties with brief sketches of their proofs. Also we emphasize the use of the operational form of polynomials, which is often the key to get closed-form results. In Section III we introduce the general Gaussian unitary and recall that it is obtained as a cascade combination of a displacement, a rotation, and a squeezing.MaRh90 In Section IV we obtain the Fock representation of the general noisy Gaussian state interpreted as the result of the application of a general Gaussian unitary to a thermal state. The derivation exploits in a natural way the HL properties discussed in Section II. The final result, expressed by a simple matrix form, depends on the parameters of the Gaussian unitary, namely, the displacement amount $\alpha$ and the squeeze amount $z$, and on the average number of the photons $\mathcal{N}$ in the thermal noise. In Section V special cases are obtained to get the Fock expansion of 1) noisy squeezed states by setting $\alpha=0$, 2) noisy displaced states by setting $z=0$, and 3) pure states by setting $\mathcal{N}=0$. Finally, in Section VI we will use the Fock expansion to evaluate the probability distribution of the photon number.

We conclude by remarking that results available in the literature, obtained by alternative methods, appear to be very different from our results, but we have checked that they are in perfect agreement with them.

II Hermite-like polynomials

Hermite polynomials with many indices and many variables have been studied since the nineteenth century. Hermite himself, in his original proposal,Herm64 introduced orthogonal polynomials with many indices and many variables. Later, Appell and Kampé de Fériet dedicated to this topic a monographic volume,Appe26 where their relevant properties have been studied in depth.

The study and the application of these families of polynomials has been revived in more recent times within different contexts, either in pure mathematics and in applications. The reasons of this interest is either because they are suited to describe physical phenomena, e.g., diffusion problems,but also because their embedding with methods of operational nature has provided new points of view on the theory of special functions and of their generalization.

II.1 Hermite-Kampé de Fériet polynomials

The two variable version of the Hermite polynomials,

[TABLE]

have been introduced in Ref. [16] and will be referred as Hermite–Kampé de Fériet (briefly H-KdF) polynomials. Since they are solutions of the heat differential equation, they are also called heat polynomials.

In the present paper wide use will be made of the operational approach, whose simplest result is given by the following shift transformation

[TABLE]

(which holds true for any function $g(x)$ admitting Taylor expansion). Then it is worth to introduce the operational definition of the H-KdF polynomials

[TABLE]

Indeed, after expanding the exponential as

[TABLE]

and, on account of the fact that

[TABLE]

for $r=0,1,\ldots,\lfloor n/2\rfloor$ while $\partial_{x}^{2r}x^{n}=0$ otherwise, one obtains

[TABLE]

Also the generating function of the H-KdF polynomials follows

[TABLE]

Finally, from the operational definition (6) a quasi-monomial property follows, namely,

[TABLE]

A result of a crucial importance in our paper is the following Mehler type addition formula

[TABLE]

Note that, by using the operational definition (3), $G(x,y;z,u|t)$ can be recast in the form

[TABLE]

Then, applying the well-known Gauss–Weierstrass transformBilo62

[TABLE]

a simple algebra yields (9).

II.2 Two-indices Hermite-Kampé de Fériet

polynomials

Let us now go a step further, by introducing the two-indices H-KdF polynomials

[TABLE]

Their operational definition is

[TABLE]

as it follows from

[TABLE]

where use has been made of the quasi-monomial property (8) of the H-KdF polynomials.

Moreover (13) gives also the generating function of the two-indices H-KdF polynomial as

[TABLE]

Note that the operational form (13) has an ambiguity in the degenerate case $x=z=0$. To overcome this ambiguity we can use the so called incomplete Hermite polynomials

[TABLE]

and referred as even ($\epsilon=0$) or odd ($\epsilon=1$). The associated operational form can be guessed from Eq. (13) and reads

[TABLE]

In this section we have provided so far the main properties of the HL polynomials. The underlying technicalities will be exploited in the forthcoming part of the paper.

III Gaussian unitaries

III.1 Definitions

A Gaussian unitary (defined as a unitary operator transforming Gaussian states into Gaussian states) can be represented in terms of three fundamental unitaries, namely a displacement operator

[TABLE]

a rotation operator

[TABLE]

and a squeezing operator

[TABLE]

where $a$ is the annihilator operator and $a^{\dagger}$ is the creation operator. In factMaRh90 ; Cari15 the most general Gaussian unitary is given by the combination of three fundamental Gaussian unitaries $D(\alpha)$, $S(z)$, and $R(\phi)$, cascaded in any arbitrary order. Without restriction, we will refer to the cascade $U=S(z)\,D(\alpha)\,R(\phi)$, because the other combinations can be easily obtained by simple transformation of the parameters.MaRh90 Under this assumption the specification of a Gaussian unitary is provided by a triple of complex parameters $(\alpha,\phi,z)$, which we call bosonic parameters.

III.2 Fock representation of a general Gaussian unitary

Even though different approaches have been used for the evaluation of the Fock coefficients of a general Gaussian state, we prefer to use a direct approach, starting from the normal ordering of the unitary operator $U$, namely,

[TABLE]

where

[TABLE]

[TABLE]

The parameters $T$ and $S$ are obtained from the squeeze parameter $z=r\,e^{i\theta}$ as $T=e^{i\theta}\tanh\,r$ and $S={\rm sech}\,r$. The factorization (22) is a particularization of the normal ordering for a general multi-mode Gaussian unitary, obtained by Ma and RhodesMaRh90 (for the single mode see also Fisher et al.Fish84 ).

The final result is simply expressed in terms of a two-indices H-KdF polynomial.

Proposition 1. The Fock coefficients of the Gaussian unitary $U=S(z)D(\alpha)R(\phi)$ are given by

[TABLE]

where

[TABLE]

are expressed in terms of the bosonic parameters.

This is the first result that states the presence of a Hermite–like polynomial. The proof of the proposition is given in Appendix A.

III.3 Particular cases

From the general result of Proposition 1 one can obtain the Fock coefficients of fundamental unitaries. To get the particular cases one has to take into account the following degenerate forms of the H KdF polynomials

[TABLE]

For a rotation we have to let

[TABLE]

to get

[TABLE]

For a displacement we have to let

[TABLE]

to get

[TABLE]

For a squeezing we have to let

[TABLE]

to get

[TABLE]

We leave ir to the reader to develop the consequent simplifications after use of Eqs. (27) and (30). Note in particular that for the displacement $U_{m,n}$ can be finally expressed through the generalized Laguerre polynomial $L^{(m-n)}_{n}(x)$.

IV Fock representation of a noisy Gaussian state

A noisy Gaussian state may be assumed as generated by applying a unitary operator to a thermal noise state as depicted in Fig. 1, so that it may be defined by specifying the thermal state and the generating Gaussian unitary. The thermal state is given by

[TABLE]

with geometrical Fock coefficients

[TABLE]

where $|n\rangle$ are the Fock states and $\mathcal{N}$ is the average number of photons in $\rho_{\rm th}$.

The most general noisy Gaussian state $\rho$ is obtained by the application of the most general Gaussian unitary to the thermal state $\rho_{\rm th}$, that is,

[TABLE]

Our target is the the evaluation of the Fock coefficients of the density operator $\rho$, defined by

[TABLE]

A slight simplification arises if one observes that the rotation operator $R(\phi)$ does not modify the number states $|n\rangle$, since $e^{i\phi a^{\dagger}a}|n\rangle=e^{in\phi}|n\rangle$ differs from $|n\rangle$ only for the inessential phase $e^{in\phi}$. As a consequence $R(\phi)\rho_{\rm th}R^{\dagger}(\phi)=\rho_{\rm th}$ and we are justified to ignore the effect of the rotation operator and to put $\phi=0$ in the following. Then $\rho$ and $\rho_{mn}$ are specified by the displacement parameter $\alpha$, the squeeze parameter $z=re^{i\theta}$, and the noise parameter $\mathcal{N}$.

IV.1 An infinite series representation

The Fock representation of the noisy density operator $\rho=U\rho_{\rm th}U^{\dagger}$ is

[TABLE]

By applying the result of Proposition 1 one gets

[TABLE]

with

[TABLE]

Then using the definition (12) of the two-indexes H-KdF polynomials gives

[TABLE]

To simplify the series we note that the range of the indexes may be rewritten as

[TABLE]

so that (38) may be rearranged as

[TABLE]

with

[TABLE]

In conclusion

[TABLE]

In this formulation the Fock coefficients are expressed through the series (69). The derivation of the relevant closed forms will be considered in the next subsections.

IV.2 Closed–form solution

According to the recurrence property under derivative given in (8) of the H-KdF polynomials yields

[TABLE]

where

[TABLE]

Then we can apply to (44) the Mehler type identity (9) to get

[TABLE]

where

[TABLE]

Note that

[TABLE]

belongs to the range $[0,1)$ so that $\sqrt{L}$ is real. We let

[TABLE]

and use the following lemma:

*Lemma.*The multiple mixed derivatives of a quadratic exponential are given by

[TABLE]

where $H_{m,n}(x,y;z,u|X)$ is the two–indices H-KdF polynomial defined by (12).

The proof is given in Appendix B.

Using the lemma gives the mixed derivative in (43), namely,

[TABLE]

with $a=LY^{2}u^{*}$ and $b=LY$.

IV.3 Final result

Combining the above results gives:

Proposition 2. The Fock coefficients of the general Gaussian state $\rho$ are given by

[TABLE]

where

[TABLE]

The coefficients appearing in these formulas are related to the parameters $\alpha$ and $z=re^{i\theta}$ of the Gaussian unitary and to the average photon number $\mathcal{N}$ of the thermal noise by the relations

[TABLE]

The Fock coefficients (51) may be collected into a matrix $\rho$ with infinite dimension, namely,

[TABLE]

where the matrix ${\bf K}=[K_{m,r}]$ and ${\bf W}=[W_{r,s}]$ are defined by (68) and (69), respectively. The matrix ${\bf K}$ is lower triangular (and ${\bf K}^{\dagger}$ is upper triangular) because $H_{m-r}(x,y)$ vanishes for $r>m$. Moreover, ${\bf W}$ is Hermitian and positive semidefinite.

Eq. (56) represents the main result of the paper. Note the very compact form notwithstanding the several variables involved in the theory. All the factors are expressed in terms of the two bosonic parameters $\alpha$ and $z=re^{i\theta}$, and of the thermal noise $\mathcal{N}$.

IV.4 Numerical results

We recall that the diagonal entries $\rho_{m,m}$ give the probability distribution of the photon number present in the state described by the noisy Gaussian state $\rho$. We have evaluated $\rho_{m,m}$ numerically for values of the bosonic parameters $\alpha$, $r$, $\theta$ and of the thermal noise $\mathcal{N}$. The results are illustrated in the following figures. The figures on the left show $\rho_{m,n}$ in the range $0\leq m\leq 6$, while the figures on the right show $\rho_{m,m}$ in the range $0\leq m\leq 50$. In the captions, the matrices on the left give the values of $\alpha,r,\theta,\mathcal{N},m_{max}$, while the matrix on the right gives the partial trace $\sum_{m=0}^{m_{\max}}\;\rho_{m,n}$.

In Fig.2 the plot refers to $r=0.5$, $\theta=0.5$, $\mathcal{N}=0.5$, and four values of $\alpha$.

In Fig.3 the plot refers to $\alpha=1+0.3i$, $\theta=0.5$, $\mathcal{N}=0.5$, and four values of $r$.

In Fig.4 the plot refers to $\alpha=1+0.3i$, $r=1$, $\mathcal{N}=0.5$, and four values of $\theta$.

In Fig.5 the plot refers to $\alpha=1+0.3i$, $r=1$, $\theta=0.5$, and four values of $\mathcal{N}$.

V Particular cases

The theory developed here for a general noisy Gaussian state, leading to the matrix form \mbox{\bm{\rho}}=J{\bf K}{\bf W}{\bf K}^{\dagger}, can be particularized to specific cases: 1) noisy displaced states, 2) noisy squeezed states, 3) pure Gaussian states. The particularization is similar to the one seen in Section III–C for the fundamental Gaussian unitaries and is based on the degenerate forms of the H-KdF polynomials given by Eqs. (27) and (30).

In Section II-B we noted that the operative form of the polynomial, which led to the general closed-form result, has an ambiguity in the degenerate cases and we gave an alternative form. However, we have verified that the general result of Proposition 2 perfectly holds also in the degenerate cases, so that the alternative form was not necessary.

We first develop a simple check. In the absence of displacement ($\alpha=0$) and squeezing ($z=0$) we have to obtain the Fock representation of the thermal state. In fact

[TABLE]

and

[TABLE]

in agreement with (31).

V.1 Noisy displaced states

By neglecting the squeeze operator and, consequently, by setting $z=re^{i\theta}=0$ and using Eq. (27), one finds

[TABLE]

and the entries of the matrices ${\bf K}$ and ${\bf W}$ become

[TABLE]

The explicit result is

[TABLE]

where $\mathcal{D}_{rs}(x,z,y)$ is the polynomial

[TABLE]

Note that a closed-form result is available in the literature due to HelstromHels76 , namely, for $n\geq m$

[TABLE]

while for $m>n$ the coefficients are obtained using the Hermitian symmetry.

The expressions (59) and (61) are quite different. The reason lies on the fact that our approach is completely different from the one followed in the cited paper. However, the check, not easy, leads to the perfect agreement of the two results.

V.2 Noisy squeezed states

The Fock expansion of a noisy squeezed state results by neglecting the displacement operator and, consequently, by setting $\alpha=0$, and using Eq. (30). We find:

[TABLE]

and the entries of the matrices ${\bf K}$ and ${\bf W}$ become

[TABLE]

In a paper of P. Marian and T. MarianMari93 a correspondig result is expressed in terms of a hypergeometric function. Also in this case the difference is due to the different approach, but the agreement of the results has been checked.

V.3 Pure Gaussian states

The Fock expansion of a pure Gaussian state results by neglecting the thermal noise, and, consequently, by setting $Y=0$. We find

[TABLE]

[TABLE]

The final expression reads

[TABLE]

This result is in agreement with previous results in the literatureKim89 ; Dodo94b . SerafiniSera17 evaluates the Fock coefficients for $\alpha=0$ and $z=e^{2r}$, obtaining a result in agreement with (70).

VI Conclusions

We have tackled the problem of the Fock representation of Gaussian unitaries and Gaussian states, with their closed-form evaluation as final target. The motivation was the possibility of evaluating the performance of quantum communications in free space and optical fiber, where a synthetic form the Fock representation is necessary. We have shown the fundamental role of the Hermite-like polynomials in this topic, especially their operational form, which is the key to reach the desired closed-form result. We have considered the single mode, but the methodology could be extended to the multi–mode case. In fact, the normal ordering of the Gaussian unitaries, which is the starting point of our derivation, is available also in the muti–mode case,MaRh90 and also the fact that exponential factors in the normal ordering have the same structure as the generating function of the Hermite-like polynomials holds also for the multimode. Of course, the complexity of the development increases dramatically with the order of the mode.

Appendix A Proof of Proposition 1.

Starting from (33) we get

[TABLE]

We compute separately the coefficients $B_{m,r}$, $C_{r,q}$, and $F_{q,n}$. A straightforward application of the properties of the number states enables us to obtain

[TABLE]

Since $F(a)$ may be expressed as a generating function of the H-KdF polynomials, namely, $F(a)=e^{ua^{2}+za}=h(z,u,a)$, one gets

[TABLE]

for $0\leq q\leq n$, while $F_{qn}=0$ otherwise. Similarly,

[TABLE]

for $0\leq r\leq m$, while $B_{mr}=0$ otherwise. Substituting these results in (71) yields

[TABLE]

and the claim is proved by virtue of (12).

Appendix B Proof of Lemma

Multiplying both sides of (49) by $\frac{u^{m}v^{n}}{m!n!}$ and summing over the indices gives

[TABLE]

and

[TABLE]

where the shift transformation (2) and the generating function (77) of the two indexes H-KdF polynomials are used.