Some results for \(q\)-functions of many variables
Summary: We will give some new formulas and expansions for \(q\)-Appell-, \(q\)-Lauricella- and \(q\)-Kampé de Fériet functions. This is an area which has been explored by Jackson, R. P. Agarwal, W. A. Al-Salam, Andrews, Kalnins, Miller, Manocha, Jain, Srivastava, and Van der Jeugt. The multiple \(q\)-hypergeometric functions are defined by the \(q\)-shifted factorial and the tilde operator. By a method invented by the author, which also involves the Ward-Alsalam \(q\)-addition and the Jackson-Hahn \(q\)-addition, we are able to find \(q\)-analogues of corresponding formulas for the multiple hypergeometric case. In the process we give a new definition of \(q\)-hypergeometric series, illustrated by some examples, which elucidate the integration property of \(q\)-calculus, and helps to try to make a first systematic attempt to find summation- and reduction theorems for multiple basic series. The Jackson \(\Gamma_q\)-function will be frequently used. Two definitions of a generalized \(q\)-Kampé de Fériet function, in the spirit of \textit{H. M. Srivastava} and \textit{P. W. Karlsson} [Multiple Gaussian hypergeometric series (1985; Zbl 0552.33001)], which are symmetric in the variables, and which allow \(q\) to be a vector are given. Various connections of transformation formulas for multiple \(q\)-hypergeometric series with Lie algebras and finite groups known from the literature are cited.
scientific article; zbMATH DE number 5121478
scientific article