Ramanujan’s master theorem for symmetric cones

Hongming Ding, Kenneth I. Gross and Donald Richards

Vol. 175 (1996), No. 2, 447–490

DOI: 10.2140/pjm.1996.175.447

Abstract

The Master Theorem of Ramanujan (1913), so named because of its centrality in much of Ramanujan’s work on definite integrals, hypergeometric functions, and series expansions, relates coefficients in the Taylor’s expansion of a function to the Mellin transform of the function over the interval (0,∞). In this paper we extend the setting of this classical theorem to apply to spherical series and spherical transforms on symmetric cones (also known as domains of positivity). To illustrate the range of applications of this theorem we obtain higher dimensional analogues of Carlson’s uniqueness theorem for holomorphic functions, Newton’s interpolation formula, and Mellin-Barnes integrals for certain hypergeometric functions.

Mathematical Subject Classification 2000

Primary: 43A90

Secondary: 17C50, 33C50

Milestones

Received: 15 February 1994

Published: 1 October 1996