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.

We study integration over functions on superspaces. These functions are invariant under a transformation which maps the whole superspace onto the part of the superspace which only comprises purely commuting variables. We get a compact expression for the differential operator with respect to the commuting variables which results from Berezin integration over all Grassmann variables. Also, we derive Cauchy-like integral theorems for invariant functions on supervectors and symmetric supermatrices. This extends theorems partly derived by other authors. As a physical application, we calculate the generating function of the one-point correlation function in random matrix theory. Furthermore, we give another derivation of supermatrix Bessel functions for U (k 1 /k 2 ).

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We extend the recursion formula for matrix Bessel functions, which we obtained previously, to superspace. It is sufficient to do this for the unitary orthosymplectic supergroup. By direct computations, we show that fairly explicit results can be obtained, at least up to dimension 8ϫ8 for the supermatrices. Since we introduce a new technique, we discuss various of its aspects in some detail.