On certain extensions of Ramanujan's Master Theorem and their applications

TL;DR

This paper extends Ramanujan's Master Theorem using k-gamma and p-k gamma functions, introduces a Mellin double integral, and applies these to evaluate various definite integrals.

Contribution

It presents novel extensions of Ramanujan's Master Theorem involving special gamma functions and establishes a new Mellin double integral with applications.

Findings

Extended Ramanujan's Master Theorem using k-gamma functions

Established a Mellin type double integral and its corollaries

Applied results to evaluate diverse definite integrals

Abstract

S. Ramanujan introduced a technique in 1913 for providing analytic expressions for certain Mellin-type integrals which is now known as Ramanujan's Master Theorem. This technique was communicated through his "Quarterly Reports" and has a wide range of applicability in calculating the values of certain definite integrals. In this paper, we have presented some extensions of Ramanujan's Master Theorem that arise from the k-gamma function and the p-k gamma function. Furthermore, we have established a Mellin type double integral along with its corollaries. Some applications are established through the evaluation of a variety of definite integrals.