The Ramanujan master theorem and its implications for special functions

TL;DR

This paper explores extensions of Ramanujan's master theorem using umbral methods, impacting special functions by deriving new integral formulas and offering a unified approach to related problems.

Contribution

It introduces a novel umbral-based extension of Ramanujan's theorem, providing new integral formulas for Bessel functions and a unified framework for exponential and Gaussian integrals.

Findings

Derived new integral formulas for Bessel functions.

Unified treatment of exponential and Gaussian integrals.

Extended Ramanujan's theorem using umbral methods.

Abstract

We study a number of possible extensions of the Ramanujan master theorem, which is formulated here by using methods of Umbral nature. We discuss the implications of the procedure for the theory of special functions, like the derivation of formulae concerning the integrals of products of families of Bessel functions and the successive derivatives of Bessel type functions. We stress also that the procedure we propose allows a unified treatment of many problems appearing in applications, which can formally be reduced to the evaluation of exponential- or Gaussian-like integrals.