On the Triple Lauricella–Horn–Karlsson q-Hypergeometric Functions

Thomas Ernst

doi:10.3390/axioms9030093

Abstract

The Horn–Karlsson approach to find convergence regions is applied to find convergence regions for triple q-hypergeometric functions. It turns out that the convergence regions are significantly increased in the q-case; just as for q-Appell and q-Lauricella functions, additions are replaced by Ward q-additions. Mostly referring to Krishna Srivastava 1956, we give q-integral representations for these functions.

Highlights

This is part of a series of papers about q-integral representations of q-hypergeometric functions
The first paper [1] was about q-hypergeometric transformations involving q-integrals
In [3], Eulerian q-integrals for single and multiple q-hypergeometric series were found. This subject is by no means exhausted, and in the same proceedings, [4], concise proofs for q-analogues of Eulerian integral formulas for general q-hypergeometric functions corresponding to Erdélyi, and for two of Srivastavas triple hypergeometric functions were given

Summary

Introduction

This is part of a series of papers about q-integral representations of q-hypergeometric functions. In [3], Eulerian q-integrals for single and multiple q-hypergeometric series were found This subject is by no means exhausted, and in the same proceedings, [4], concise proofs for q-analogues of Eulerian integral formulas for general q-hypergeometric functions corresponding to Erdélyi, and for two of Srivastavas triple hypergeometric functions were given. The history of the subject in this article started in 1889 when Horn [6] investigated the domain of convergence for double and triple q-hypergeometric functions. He invented an ingenious geometric construction with five sets of convergence regions in three dimensions which was successfully used by Karlsson [7] in 1974 to explicitly state the convergence regions for the known functions of three variables.

Definitions

Discussion

Conclusions