Some closed-form evaluations of multiple hypergeometric and \(q\)-hypergeometric series (Q1776824)

Starting from a well-known summation formula of Carlitz for multiple hypergeometric functions, the authors show that some fairly general analytical tools and techniques can be applied in order to obtain summation, transformation and reduction formulas for multiple hypergeometric and multiple basic hypergeometric series. They derive, in this paper, several closed-form expressions for various families of multiple hypergeometric series, including Carlitz summation formula, by applying some known reduction formulas for multi-variable hypergeometric functions. They also consider \(q\)-extensions of Carlitz's multiple sums as well as some closely-related transformation and reduction formulas for multi-variable \(q\)-hypergeometric functions. A number of multiple \(q\)-series identities, which are developed in this paper, are observed to be potentially useful in the related problems involving closed-form evaluations of multi-variable \(q\)-hypergeometric functions.