Type-(I,II) Interpolations and some asymptotic expansions using Ramanujan's master theorem

TL;DR

This paper extends Ramanujan's master theorem using Mellin transform techniques, applies it to special functions like the Riesz function, and derives new asymptotic expansions.

Contribution

It introduces extended versions of Ramanujan's master theorem and demonstrates their application to special functions and asymptotic analysis.

Findings

Derived extended Ramanujan's master theorem variants

Applied the theorem to Riesz and binomial functions

Obtained new asymptotic expansions

Abstract

The theory of Mellin transform is an incredibly useful tool in evaluating some of the well known results for the zeta function. Ramanujan in his quarterly reports \cite{1} gave a theorem for Mellin transform which is now known as Ramanujan's master theorem \cite{2}. In this paper, we have derived some extended versions of Ramanujan's master theorem based on our previous results \cite{3} and applied them to some special functions such as known as the Riesz function $R(z)$ and generalized binomial function. Some asymptotic expansions using extended Ramanujan's master theorem are also derived.