Communications of the
Korean Mathematical Society CKMS

In this paper we extend the two $q$-additions with powers in the umbrae, define a $q$-multinomial-coefficient, which implies a vector version of the $q$-binomial theorem, and an arbitrary complex power of a JHC power series is shown to be equivalent to a special case of the first $q$-Lauricella function. We then present several $q$-analogues of hypergeometric integral formulas from the two books by Exton and the paper by Choi and Rathie. We also find multiple $q$-analogues of hypergeometric integral formulas from the recent paper by Kim. Finally, we prove several multiple $q$-hypergeometric integral formulas emanating from a paper by Koschmieder, which are special cases of more general formulas by Exton.

Keywords: Eulerian $q$-integral, $q$-multinomial-coefficient, Srivastava $\triangle$ notation, multiple $q$-beta function

MSC numbers: Primary 33D05, 33D15; Secondary 33C65, 33D60