Special Functions on Pos_n

Abstract

Classical functions such as the gamma function and the Bessel function have analogues on symmetric spaces, as do certain classical integral transforms. For the generalization of gamma function to Posn, the idea goes back to Siegel [Sie 35], and for the Bessel function it goes back to Bochner [Boc 52], Herz [Her 55] and Selberg [Sel 56]. We shall give further bibliographical comments later. Among the integral transforms is the generalized Mellin transform. Cf. Gindikin [Gin 64], who provides a beautiful survey of special functions on spaces like, but more general than Posn. Thus large portions of harmonic analysis, as well as the theory of Dirichlet and Bessel series carries over to such spaces. Here we are concerned with the most standard of all symmetric spaces, the space Posn of symmetric positive definite real matrices. As the reader will see, one replaces the invariant measure dy/y on the multiplicative group by the measure |Y |^-(n+1)/2dμ_euc(Y) on the space of positive definite matrices. We develop systematically the theory of some special functions on this space, namely, the gamma and K-Bessel functions, as a prototype of other special functions, and also prototype of more general symmetric spaces. No matter what, it is useful to have tabulated the formulas in this special case, for various applications.