Ramanujan's Master theorem for the hypergeometric Fourier transform on root systems

TL;DR

This paper extends Ramanujan's Master theorem to the hypergeometric Fourier transform on root systems, providing a new interpolation formula that generalizes previous results for symmetric spaces.

Contribution

It proves an analogue of Ramanujan's Master theorem for hypergeometric Fourier transforms on root systems, broadening the scope of the original theorem.

Findings

Establishes a new interpolation formula for hypergeometric Fourier transforms.

Generalizes previous results from symmetric spaces to arbitrary root systems.

Provides theoretical foundation for further applications in harmonic analysis.

Abstract

Ramanujan's Master theorem states that, under suitable conditions, the Mellin transform of an alternating power series provides an interpolation formula for the coefficients of this series. Ramanujan applied this theorem to compute several definite integrals and power series, which explains why it is referred to as the "Master Theorem". In this paper we prove an analogue of Ramanujan's Master theorem for the hypergeometric Fourier transform on root systems. This theorem generalizes to arbitrary positive multiplicity functions the results previously proven by the same authors for the spherical Fourier transform on semisimple Riemannian symmetric spaces.