Bessel functions on \(P_ n\) (Q2266095)

This paper considers analogues of the K-Bessel function associated to the general linear group. Such functions have been studied for applications in multivariate statistics and in the theory of automorphic forms for GL(n,\({\mathbb{Z}})\). References are \textit{C. S. Herz}, Ann. Math., II. Ser. 61, 474-523 (1955; Zbl 0066.320) and the reviewer's, Can. Math. Bull. (to appear). The main result is an integral formula which says that the definition which gives a rather easy analytic continuation in the complex parameters is the same as the definition which gives an obvious partial differential equation in the symmetric space variable. It is also shown that one can evaluate a higher dimensional Mellin transform of the K- Bessel function. And a useful inductive formula is obtained - a result which was used in work by Imai and the reviewer on Fourier expansions of Eisenstein series.