Recursive construction for a class of radial functions. I. Ordinary space

Abstract

A class of spherical functions is studied which can be viewed as the matrix generalization of Bessel functions. We derive a recursive structure for these functions. We show that they are only special cases of more general radial functions which also have a, properly generalized, recursive structure. Some explicit results are worked out.

Interpretations of some parameter dependent generalizations of classical matrix ensembles

“…As a consequence we can specify the sought eigenvalue distribution. In the case that all eigenvalues of A are distinct (or doubly degenerate in the case of x complex), the eigenvalues of M can be interpreted as so called radial Gelfand-Tzetlin coordinates introduced by Guhr and Kohler [22]. Further, as shown in Appendix C, this observation and Corollary 1 can be used to rederive a recursion formula obtained in [22] for certain matrix Bessel functions.…”

“…A simple calculation (see [22]) shows that the eigenvalues of (C.2) coincide with the non-zero eigenvalues of (C.1). Consequently the second exponential in the integrand only depends on U N , so if we decompose…”

“…In [22] the non-zero eigenvalues of this projection are termed the radial Gelfand-Tzeltin coordinates. Let A denote the N × N − 1 matrix with 1's down its diagonal and 0's elsewhere.…”

Derivation of the supersymmetric Harish-Chandra integral for UOSp(k1/2k2)

“…This method has not been applied yet to work out HarishChandra integrals for the ordinary orthogonal or unitary symplectic or the supersymmetric unitary orthosymplectic supergroup. However, it has been employed for Gelfand's spherical functions 21,22,23,24 .…”

Angular Gelfand–Tzetlin coordinates for the supergroup UOSp(k1/2k2)

“…By that we mean, that they never leave the space of the group and its algebra. In previous contributions 9, 10,11,12 , we constructed radial Gelfand-Tzetlin coordinates to study certain types of group integrals. These radial Gelfand-Tzetlin coordinates are capable of mapping the integral over a group onto integrals over the radial part of a different symmetric space.…”