A New q-Hypergeometric Symbolic Calculus in the Spirit of Horn, Borngässer, Debiard and Gaveau

Thomas Ernst

doi:10.3390/axioms11020064

Abstract

The purpose of this article is to introduce a new complete multiple q-hypergeometric symbolic calculus, which leads to q-Euler integrals and a very similar canonical system of q-difference equations for multiple q-hypergeometric functions. q-analogues of recurrence formulas in Horns paper and Borngässers thesis lead to a more exact way to find these Frobenius solutions. To find the right formulas, the parameters in q-shifted factorials can be changed to negative integers, which give no extra q-factors. In proving these q-formulas, in the limit q→1 we obtain versions of the paper by Debiard and Gaveau for the solution of differential or q-difference equations. The paper is also a correction of some of the statements in the paper by Debiard and Gaveau, e.g., the Euler integrals and other solutions to differential equations for Appell functions, also without references to page numbers in the standard work of Appell and Kampé de Fériet. Sometimes the q-binomial theorem is used to simplify q-integral formulas. By the Horn method, we find another solution to the Appell Φ1 function partial differential equation, which was not mentioned in the thesis by Le Vavasseur 1893.

Highlights

We refer to our standard work [1] and to the paper on multiple q-hypergeometric functions [2]
The pathbreaking paper [3] by Debiard and Gaveau on a new umbral calculus led to the automatic solutions of differential equations for multiple hypergeometric functions according to Frobenius and Horn
Our method leads to more direct computation of the other solutions of Appell differential and similar differential equations than the papers by Horn and Borngässer

Summary

Introduction

We refer to our standard work [1] and to the paper on multiple q-hypergeometric functions [2]. The pathbreaking paper [3] by Debiard and Gaveau on a new umbral calculus led to the automatic solutions of differential equations for multiple hypergeometric functions according to Frobenius and Horn. We generalize this method to the q-case and slightly change the notation for a better overview. The exponent in method of Frobenius is changed from α to λ and the. Euler operator x d dx is changed to θ(q) as in [1]. Our umbral calculus means that a θq,1 ∨ θq, before a double power series is replaced by the exponents of x ∨ y. The same goes for additive arguments in the

Objectives

Discussion

Conclusion