Convergence aspects for generalizations of \(q\)-hypergeometric functions (Q2352966)

Summary: In an earlier paper [``On certain generalizations of \(q\)-hypergeometric functions of two variables'', Int. J. Math. Comput. 16, 1--27 (2012)], 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 \textit{P. W. Karlsson} [On certain generalizations of hypergeometric functions of two variables. Lyngby, Denmark: Physics Laboratory II, The Technical University of Denmark (1976; Zbl 0328.33004)]. The \(\Gamma_q\)-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.