Formal (q-)Euler integrals over the unit hypercube and over triangles in higher dimensions for multiple (q-)hypergeometric functions

Abstract

This article contains both multiple hypergeometric functions and cor- responding q-analogues. First we present integral expressions for multi- ple hypergeometric functions over the unit hypercube and over triangles in higher dimensions. Then we extend these integrals to the q-case by using the q-real number R⊞q . The q-binomial theorem, the q-beta integral and their generalizations to higher dimensions are used in the proofs. Also confluent forms with the Euler q-exponentoial function are proved. Reduction for- mulas for Kampe ́ de Fe ́riet functions are proved by using Euler integrals, Beta integrals and hypergeometric transformations. Finally, Euler integral representations for Horn functions and q-Euler integral representations of q-Kampe ́ de Fe ́riet functions are proved.