In this paper we give an extension of -Pfaff-Saalschütz formula by means of Andrews-Askey integral. Applications of the extension are also given, which include an extension of -Chu-Vandermonde convolution formula and some other -identities.
We study -integral representations of the -gamma and the -beta functions. As an application of these integral representations, we obtain a simple conceptual proof of a family of identities for Jacobi triple product, including Jacobi's identity, and of Ramanujan's formula for the bilateral hypergeometric series.
This paper is divided in two parts. In the first part we study a convergent -analog of the divergent Euler series, with , and we show how the Borel sum of a generic Gevrey formal solution to a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of a corresponding -difference equation. In the second part, we work under the assumption . In this case, at least four different -Borel sums of a divergent power series solution of an irregular singular...