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

2012/10/31 by Gestur Ólafsson, Olafsson, Gestur, Angela Pasquale +1
Mathematics · #33CC67 #43A32 #43A90 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1211.0024

openalex publication_date 2012/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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