Ramanujan's master theorem for symmetric cones
Ramanujan's Master Theorem from 1913, which relates coefficients in the Taylor expansion of a function to the Mellin transform of the function over the interval \((0,\infty)\), is generalized to symmetric cones in the paper under review. An outline of the paper follows. The first part is mainly expository, dealing at some length with the necessary background material on symmetric cones, Jordan algebras, and harmonic analysis on symmetric cones, including Pochhammer symbol, gamma function, zonal polynomials, spherical functions, and Plancherel's formula. Then after introducing Hardy classes, the authors prove the Master Theorem. The classical case is first considered and then the rather long proof for the symmetrical cone case follows. In the final sections the authors study further subjects connected with the theorem: Newton's interpolation formula, generalized binomial coefficients, Mellin-Barnes integrals, and hypergeometric series.
scientific article; zbMATH DE number 1026457