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Ramanujan’s master theorem for symmetric cones. (English) Zbl 0880.33013
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 $\left(0,\infty \right)$, 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.

##### MSC:
 33C80 Connections of hypergeometric functions with groups and algebras 44A15 Special transforms (Legendre, Hilbert, etc.)