Some families of hypergeometric generating functions associated with multiple series transformations

doi:10.1016/S0096-3003(02)00390-9

Applied Mathematics and Computation

Volume 144, Issue 1, 20 November 2003, Pages 27-59

https://doi.org/10.1016/S0096-3003(02)00390-9 Get rights and content

Abstract

A simple multiple-series identity is applied here with a view to deriving, in a systematic and unified manner, numerous families of generating functions for certain interesting classes of generalized hypergeometric polynomials in one, two, and more variables. Relevant connections of many of these families of generating functions with various known results on this subject are also discussed. The use of double-, triple-, and multiple-series analogues of the familiar Bailey transform, which were invoked in many of the earlier works cited here, is completely avoided in this investigation.

Introduction

For suitably bounded complex sequences {α_n}_n=0^∞, {δ_n}_n=0^∞, {u_n}_n=0^∞, and {v_n}_n=0^∞, if we let $β_{n} :=∑_{j=0}^{n} α_{j} u_{n−j} v_{n+j}$ and $γ_{n} :=∑_{j=n}^{∞} δ_{j} u_{j−n} v_{j+n} =∑_{j=0}^{∞} δ_{n+j} u_{j} v_{2n+j},$ then it is easily seen that $∑_{n=0}^{∞} α_{n} γ_{n} =∑_{n=0}^{∞} β_{n} δ_{n},$ provided, of course, that the inversion of the order of summation is permissible by absolute convergence of the series involved.

The series transformation (1.3) was given explicitly by Bailey [1], although the germ of this idea can be found in the work of Abel in the earlier part of the nineteenth century (see also Slater [17, p. 58 et seq.]).

Some fairly straightforward double- as well as multiple-series analogues of Bailey’s series transformation (1.3) were considered by Exton [4, pp. 33 and 139], who (just as Slater [17, p. 60] did) applied these analogous series transformations in order to deduce several interesting generating functions involving generalized hypergeometric polynomials in one, two, and more variables (cf. [4]; see also [5], [6], [7], [8], [9], [10], [11]). The main object of this paper is first to present a simple multiple-series identity and to show how readily (and systematically) these and many other hypergeometric generating functions can be derived from this multiple-series identity without using Bailey’s series transformation (1.3) and its aforementioned analogues. We also record a rather straightforward multiple-series extension of Bailey’s series transformation (1.3), which does indeed contain the aforementioned multiple-series transformation of Exton [4, p. 139].

Section snippets

A multiple-series identity

Suppose that ${Θ(n_{1},…,n_{r})}, {Φ(n_{1},…,n_{r})}, {Ψ(n_{1},…,n_{r})}, and {Ω(n_{1},…,n_{r})} (n_{j} ∈ N_{0} := N ∪{0}; N :={1,2,3,…}; j=1,…,r)$ are suitably bounded multiple sequences of complex numbers. Also let {ϕ(n)}_n=0^∞ denote a suitably bounded sequence of complex numbers. Then, assuming that the inversion of the order of summation is permitted by absolute convergence of the multiple series involved, we readily obtain the following general multiple-series identity: $∑_{n_{1},…,n_{r}=0}^{∞} ϕ(n_{1} +⋯+n_{r})Ω(n_{1},…,n_{r})(μ_{1})_{n_{1}} ⋯(μ_{r})_{n_{r}} t_{1}^{n_{1}} n_{1}! ⋯ t_{r}^{n_{r}} n_{r}! ∑_{k_{1}=0}^{[n_{1}/m}$

Some interesting deductions

First of all, in the multiple-series identity (2.8), we set $ϕ(n)= (λ)_{n} (ν)_{n} and ϑ(n)=ψ(n)=1 (n∈ N_{0}; λ∈ C; ν∈ C ⧹ Z_{0}^{−}; Z_{0}^{−} :={0,−1,−2,…})$ or $ϑ(n)= (λ)_{n} (ν)_{n} and ϕ(n)=ψ(n)=1 (n∈ N_{0}; λ∈ C; ν∈ C ⧹ Z_{0}^{−}),$ and apply the Pfaff–Kummer transformation for the Gauss hypergeometric function $_{2} F_{1}$ (cf., e.g., [2, p. 300, Eq. 9.4 (79)]): $_{2} F_{1} (α,β;γ;z)=(1−z)^{−α}_{2} F_{1} α,γ−β;γ; z z−1 (α,β∈ C; γ∈ C ⧹ Z_{0}^{−}; | arg (1−z)|⩽π−ε; 0<ε<π).$

We thus obtain the following consequences of the multiple-series identity (2.8): $∑_{n_{1},…,n_{r}=0}^{∞} (λ)_{n_{1}+⋯+n_{r}} (ν)_{n_{1}+⋯+n_{r}} (μ_{1})_{n_{1}} ⋯(μ_{r})_{n_{r}} t^{n_{1}+⋯+n_{r}} n_{1}$

Applications involving multivariable hypergeometric polynomials

By suitably specializing the sequences ${ϑ(n)}_{n=0}^{∞}, {ϕ(n)}_{n=0}^{∞}, {ψ(n)}_{n=0}^{∞}, and Φ(k_{1},…,k_{r}) (k_{j} ∈ N_{0}; j=1,…,r),$ we can apply each of the multiple-series identities of the preceding sections in order to derive the corresponding generating functions for various families of multivariable hypergeometric polynomials. In particular, if we set $m_{j} =m (j=1,…,r; m∈ N),$ make use of the familiar identity: $(λ)_{mk} =m^{mk} ∏_{j=1}^{m} λ+j−1 m_{k}$ and denote by Δ(m;λ) the array of m parameters: $λ m, λ+1 m,…, λ+m−1 m (λ∈ C; m∈ N),$ the multiple-series

Further remarks and observations

Bailey’s series transformation (1.3) as well as its aforementioned multiple-series analogues were applied extensively in many other recent works of Exton (cf., e.g., [8], [9], [10]). Most (if not all) of the single, double, and multiple hypergeometric generating functions, which were derived in this manner in each of these recent works, can be obtained by appropriately specializing one or the other of the multivariable hypergeometric generating functions considered in this paper. Just for an

Acknowledgements

The present investigation was completed during the third-named author’s visits to Tamkang University at Tamsui in April 2002 and May 2002. This work was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.

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Applied Mathematics and Computation

Some families of hypergeometric generating functions associated with multiple series transformations

Abstract

Introduction

Section snippets

A multiple-series identity

Some interesting deductions

Applications involving multivariable hypergeometric polynomials

Further remarks and observations

Acknowledgements

Identities of the Rogers–Ramanujan type

Proc. London Math. Soc. Ser. 2

Some hypergeometric polynomials associated with the Lauricella function FD of several variables. I and II

Mat. Vesnik

Beitrag zur Theorie der knonfluenten hypergeometrischen Funktionen von mehreren Veränderlichen

S.-B. Akad. Wiss. Abt. IIa Math.-Natur. Kl.

Multiple Hypergeometric Functions and Applications

A multidimensional generating function

Jñānābha

Two new multivariable generating relations

Ark. Mat.

Generating relations for Tratnik’s multivariable biorthogonal continuous Hahn polynomials

J. Math. Phys.

A multiple generating function suggested by an extension of a theorem by Bailey

Pure Appl. Math. Sci.

New generating relations for products of two Laguerre polynomials

Indian J. Pure Appl. Math.

Generating functions for a quadruple hypergeometric polynomial with SL symmetry

Indian J. Pure Appl. Math.

New generating functions for multivariate biorthogonal polynomials on the N-sphere

J. Appl. Math. Stochast. Anal.

Families of orthogonal and biorthogonal polynomials on the N-sphere

SIAM J. Math. Anal.

Some hypergeometric polynomials associated with the Lauricella function F_D of several variables. I and II