A Tribute to Richard Askey, and an Expansive View of Some of his Favorite Beta Integrals

TL;DR

This paper pays tribute to Dick Askey by re-examining his favorite Cauchy beta integral and placing it within a unifying Parseval-based framework. It revisits the one-dimensional integral and extends the idea to multidimensional settings on the space of real symmetric matrices, yielding matrix beta and Cauchy–Selberg type integrals via Fourier-Parseval techniques. The work connects classical beta and gamma functions with Wishart theory on cones and demonstrates a general methodology that leverages Parseval identities across transforms (Fourier, Mellin, Hankel) to generate broad families of integral identities. It highlights Askey's foresight in recognizing deep connections between special functions, probability (Wishart), and multivariate harmonic analysis, and sketches numerous avenues for future generalizations on symmetric cones and transform domains.

Abstract

This article represents a personal tribute to Richard Askey together with a new look at some of his favorite integrals, including the Cauchy beta integral. The article also provides some new multidimensional extensions of Cauchy's beta integral in which the domain of integration is the space of real symmetric matrices, and these multidimensional integrals are used to obtain some special cases of the Cauchy--Selberg integrals.