Abstract
In an earlier paper, we found transformation and summation formulas for 43 q-hypergeometric functions of 2n variables. The aim of the present article is to find convergence regions and a few conjectures of convergence regions for these functions based on a vector version of the Nova q-addition. These convergence regions are given in a purely formal way, extending the results of Karlsson (1976). The Γq-function and the q-binomial coefficients, which are used in the proofs, are adjusted accordingly. Furthermore, limits and special cases for the new functions, e.g., q-Lauricella functions and q-Horn functions, are pointed out.
Highlights
The standard work for multiple hypergeometric functions is [1], written by Karlsson and Srivastava.In a preprint from 1976 [2], Per Karlsson found that the restriction of multiple hypergeometric functions to an even number of variables gives a large amount of symmetry and, gives clearly defined convergence regions, integral representations, transformations and reducible cases.Based on [2] and an earlier paper [3], the aim of the present study is to present convergence regions for q-functions of 2n variables
Our philosophy is that 2n is an even number, such that the missing generalizations, or some of them at least, could be discovered by consideration of suitable hypergeometric functions, which depend on an even number of variables
We do expect such functions to Axioms 2015, 4 possess more complicated parameter systems than the four q-Lauricella functions; on the other hand, they should not be so complicated that a practical notation becomes impossible; a reasonably high symmetry will be required
Summary
The standard work for multiple hypergeometric functions is [1], written by Karlsson and Srivastava. In a preprint from 1976 [2], Per Karlsson found that the restriction of multiple hypergeometric functions to an even number of variables gives a large amount of symmetry and, gives clearly defined convergence regions, integral representations, transformations and reducible cases. To make the proofs in the current article, we use vector versions of the Γq -function, the q-binomial coefficients and the q-Stirling formula. This paper is organized as follows: we give the basic definitions of the first q-functions, together with the Nova q-addition. The following functions were defined in [3]; with the exception of I,J,L,M, capital italics denote sums, e.g., n. G2 (a, a0 ; b, b0 |q; ~x, ~y ) ha; qi|i| ha0 ; qi|j| hb; qi|j|−|i| hb0 ; qi|i|−|j|
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