Some new integral representations for the generalized hypergeometric function and their applications

Abstract

In this paper, we provide some new integral representations for the generalized hypergeometric function. These results represent a generalization of formula [15.3.1, p.558] in [Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards, Applied Mathematics Series 55, 10th printing, Washington] by Abramowitz and Stegun, extending it beyond the Gauss hypergeometric function to include the generalized hypergeometric function. As applications of these results, we present several new integral representations for Catalan number and its different generalizations are established. In the end we also provide a derivative for the generalized hypergeometric function. This provides a broader framework for applying integral representations in various mathematical contexts, particularly in applied mathematics, functional analysis, and the theory of special functions.