Evaluation of beta integrals of Ramanujan type and integral representations for bilateral hypergeometric series

TL;DR

This paper evaluates Ramanujan-type beta integrals involving gamma functions, expresses them via bilateral hypergeometric series, and provides new integral representations for these series, enhancing understanding of their summability and properties.

Contribution

It introduces new methods to evaluate Ramanujan-type integrals and derives integral representations for bilateral hypergeometric series, linking gamma functions with these series.

Findings

Expressed Ramanujan-type integrals in terms of bilateral hypergeometric series

Derived conditions under which these series are summable and integrable as beta integrals

Provided new integral representations for bilateral hypergeometric series

Abstract

In this paper we evaluate integrals of products of gamma functions of Ramanujan type in terms of bilateral hypergeometric series. In cases where the bilateral hypergeometric series are summable, then we evaluate these integral as beta integrals. In addition, we obtain integral representations for bilateral hypergeometric series.