An Operational Calculus Generalization of Ramanujan's Master Theorem

TL;DR

This paper extends Ramanujan's master theorem using operational calculus, transforming integral products into Laplace transforms, and demonstrates its consistency and applications through examples and connections to classical identities.

Contribution

It introduces a formal operational calculus extension of Ramanujan's master theorem, providing new identities and linking to Hardy and Carr's work.

Findings

Operational extension transforms integral products into Laplace transforms.

The approach is consistent with standard calculus results.

New identities are derived as corollaries.

Abstract

We give a formal extension of Ramanujan's master theorem using operational methods. The resulting identity transforms the computation of a product of integrals on the half-line to the computation of a Laplace transform. Since the identity is purely formal, we show consistency of this operational approach with various standard calculus results, followed by several examples to illustrate the power of the extension. We then briefly discuss the connection between Ramanujan's master theorem and identities of Hardy and Carr before extending the latter identities in the same way we extended Ramanujan's. Finally, we generalize our results, producing additional interesting identities as a corollary.