Some curious q-series expansions and beta integral evaluations

We deduce several curious q-series expansions by applying inverse relations to certain identities for basic hypergeometric series. After rewriting some of these expansions in terms of q-integrals, we obtain, in the limit q→ 1, some curious beta-type integral evaluations which appear to be new. Dedicated to Dick Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—15A09, 33D15, 33E20; Secondary—05A30 M. Schlosser was fully supported by an APART fellowship of the Austrian Academy of Sciences.     

q-Hypergeometric Series and Macdonald Functions

We derive a duality formula for two-row Macdonald functions by studying their relation with basic hypergeometric functions. We introduce two parameter vertex operators to construct a family of symmetric functions generalizing Hall-Littlewood functions. Their relation with Macdonald functions is governed by a very well-poised q-hypergeometric functions of type ₄₃, for which we obtain linear transformation formulas in terms of the Jacobi theta function and the q-Gamma function. The transformation formulas are then used to give the duality formula and a new formula for two-row Macdonald functions in terms of the vertex operators. The Jack polynomials are also treated accordingly.     

An Euler product transform applied to q-series

This paper introduces the concept of a D-analogue. This is a Dirichlet series analogue for the already known and well researched hypergeometric q-series, often called the basic hypergeometric series. The main result in this paper is a transform, based on an Euler product over the primes. Examples given are D-analogues of the q-binomial theorem and the q-Gauss summation. 2000 Mathematics Subject Classification Primary—11M41; Secondary—33D15, 30B50     

Transformation and reduction formulae for double q‐Clausen hypergeometric series

By means of the Sears transformations, we establish eight general transformation theorems on bivariate basic hypergeometric series. Several transformation, reduction and summation formulae on the double q‐Clausen hypergeometric series are derived as consequences. Copyright © 2007 John Wiley & Sons, Ltd.     

Reduction and transformation formulas for the Appell and related functions in two variables

In many seemingly diverse areas of applications, reduction, summation, and transformation formulas for various families of hypergeometric functions in one, two, and more variables are potentially useful, especially in situations when these hypergeometric functions are involved in solutions of mathematical, physical, and engineering problems that can be modeled by (for example) ordinary and partial differential equations. The main object of this article is to investigate a number of reductions and transformations for the Appell functions F₁,F₂,F₃, and F₄ in two variables and the corresponding (substantially more general) double‐series identities. In particular, we observe that a certain reduction formula for the Appell function F₃ derived recently by Prajapati et al., together with other related results, were obtained more than four decades earlier by Srivastava. We give a new simple derivation of the previously mentioned Srivastava's formula 12 . We also present a brief account of several other related results that are closely associated with the Appell and other higher‐order hypergeometric functions in two variables. Copyright © 2017 John Wiley & Sons, Ltd.     

Reduction Formulas for Karlsson–Minton-Type Hypergeometric Functions

We prove a master theorem for hypergeometric functions of Karlsson–Minton type, stating that a very general multilateral U(n) Karlsson–Minton-type hypergeometric series may be reduced to a finite sum. This identity contains the Karlsson–Minton summation formula and many of its known generalizations as special cases, and it also implies several Bailey-type identities for U(n) hypergeometric series, including multivariable ₁₀W₉ transformations of Denis and Gustafson and of Kajihara. Even in the one-variable case our identity is new, and even in this case its proof depends on the theory of multivariable hypergeometric series.     

一个含四个独立基的变换公式及其应用

本文利用反演的方法得到了一个四个独立基的变换公式并由此得到了几个新的基本超几何级数求和公式和超几何级数求和公式.     

LU factorizations, q = 0 limits, and p-adic interpretations of some q-hypergeometric orthogonal polynomials

For little q-Jacobi polynomials and q-Hahn polynomials we give particular q-hypergeometric series representations in which the termwise q = 0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as LU factorizations. We develop a general theory of LU factorizations related to complete systems of orthogonal polynomials with discrete orthogonality relations which admit a dual system of orthogonal polynomials. For the q = 0 orthogonal limit functions we discuss interpretations on p-adic spaces. In the little 0-Jacobi case we also discuss product formulas. Dedicated to Dick Askey on the occasion of his seventieth birthday. 2000 Mathematics Subject Classification Primary—33D45, 33D80 Work done at KdV Institute, Amsterdam and supported by NWO, project number 613.006.573.     

Some families of combinatorial and other series identities and their applications

The main object of this presentation is to show how some simple combinatorial identities can lead to several general families of combinatorial and other series identities as well as summation formulas associated with the Fox-Wright function _pΨ_q and various related generalized hypergeometric functions. At least one of the hypergeometric summation formulas, which is derived here in this manner, has already found a remarkable application in producing several interesting generalizations of the Karlsson-Minton summation formula. We also consider a number of other combinatorial series identities and rational sums which were proven, in recent works, by using different methods and techniques. We show that much more general results can be derived by means of certain summation theorems for hypergeometric series. Relevant connections of the results presented here with those in the aforementioned investigations are also considered.     

On Solutions of the q-Hypergeometric Equation with q N = 1

We consider the q-hypergeometric equation with q ^N = 1 and , , . We solve this equation on the space of functions given by a power series multiplied by a power of the logarithmic function. We prove that the subspace of solutions is two-dimensional over the field of quasi-constants. We get a basis for this space explicitly. In terms of this basis, we represent the q-hypergeometric function of the Barnes type constructed by Nishizawa and Ueno. Then we see that this function has logarithmic singularity at the origin. This is a difference between the q-hypergeometric functions with 0 < |q| < 1 and at |q| = 1.     

Extending the factorization principle to hypergeometric series of general form

It is shown that the formulas of operator factorization of hypergeometric functions obtained in the author’s previous works can be extended to hypergeometric series of the most general form. This generalization does not make the technical apparatus of the factorization method more complicated. As an example illustrating the practical effectiveness of the formulas obtained in the paper, we analyze transformation properties of the Horn seriesG ₃, whose structure is typical for general hypergeometric functions. It is shown that Erdélyi’s transformation formula relating the seriesG ₃ to the Appell functionF ₂, contains erroneous expressions in the arguments ofG ₃. The correct analog of Erdélyi’s formula is found, and some new transformations of the seriesG ₃ are presented. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 573–581, April, 2000.     

<Emphasis Type="Italic">q</Emphasis>-Hypergeometric Proofs of Polynomial Analogues of the Triple Product Identity,Lebesgue’s Identity and Euler’s Pentagonal Number Theorem

We present alternative, q-hypergeometric proofs of some polynomial analogues of classical q-series identities recently discovered by Alladi and Berkovich, and Berkovich and Garvan.Work supported by the Australian Research Council     

Some systems of multivariable orthogonal q-Racah polynomials

In 1991 Tratnik derived two systems of multivariable orthogonal Racah polynomials and considered their limit cases. q-Extensions of these systems are derived, yielding systems of multivariable orthogonal q-Racah polynomials, from which systems of multivariable orthogonal q-Hahn, dual q-Hahn, q-Krawtchouk, q-Meixner, and q-Charlier polynomials follow as special or limit cases. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D50; Secondary—33C50 Supported in part by NSERC grant #A6197.     

Euler’s constant, q-logarithms, and formulas of Ramanujan and Gosper

The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values of the q-logarithm function, with natural choice of~q. By these means, we generalize a classical formula for γ due to Ramanujan, together with Vacca’s and Gosper’s series for γ, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler’s constant. The main tools are Euler-type integrals and hypergeometric series. 2000 Mathematics Subject Classification Primary—11Y60; Secondary—11J72, 33C20, 33D15 The work of the second author is supported by an Alexander von Humboldt research fellowship Dedication: To Leonhard Euler on his 300th birthday.     

APPLICATIONS OF HYPERGEOMETRIC SUMMATION THEOREMS OF KUMMER AND DIXON INVOLVING DOUBLE SERIES

Using series iteration techniques identities and apply each of these identities in we derive a number of general double series order to deduce several hypergeometric reduction formulas involving the Srivastava-Daoust double hypergeometric function. The results presented in this article are based essentially upon the hypergeometric summation theorems of Kummer and Dixon.     

Hypergeometric series summations and π-formulas

By two Slater's hypergeometric series identities, we establish two general summation formulas with six free parameters. Specializing certain parameters in these formulas, we obtain a list of Ramanujan-type series formulas for π  , π²${\pi }^2$ and π³${\pi }^3$.     

New transformations for elliptic hypergeometric series on the root system A n

Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system A _n , with different dimensions n. We give, with a new, elementary proof, an elliptic extension of this transformation. We also obtain further Bailey-type transformations as consequences of our result, some of which are new also in the case of basic and classical hypergeometric series. 2000 Mathematics Subject Classification Primary—33D67; Secondary—11F50     

Applications of hypergeometric summation theorems of kummer and dixon involving double series

Using series iteration techniques, we derive a number of general double series identities and apply each of these identities in order to deduce several hypergeometric reduction formulas involving the Srivastava-Daoust double hypergeometric function. The results presented in this article are based essentially upon the hypergeometric summation theorems of Kummer and Dixon.     

Decomposition formulas for some triple hypergeometric functions

With the help of some techniques based upon certain inverse pairs of symbolic operators, the authors investigate several decomposition formulas associated with Srivastava's hypergeometric functions HA, HB and HC in three variables. Many operator identities involving these pairs of symbolic operators are first constructed for this purpose. By means of these operator identities, as many as 15 decomposition formulas are then found, which express the aforementioned triple hypergeometric functions in terms of such simpler functions as the products of the Gauss and Appell hypergeometric functions. Other closely-related results are also considered briefly.     

Some Cubic Summation and Transformation Formulas

q-Analogues of two cubic summation formulas that have recently caught the attention of Bill Gosper are found by first showing their connection with the q-binomial formula and then using some known transformation formulas. We also find a q-extension of a cubic transformation formula involving Gauss' hypergeometric function, which turns out to be a relation between balanced and very-well-poised 109 series.