The Asymptotic Expansion of Spherical Functions on Symmetric Cones

In [7], Genkai Zhang gives the asymptotic expansion for the spherical functions on symmetric cones. This is done to prove a central limit theorem for these spaces. The work of Zhang is a natural continuation of the work of Audrey Terras [6] (the case of the positive definite matrices of rank 2) and of the work of Donald St.P. Richards [3] (the case of the positive definite matrices of all ranks). In each case, the focus is to find the expansion of the spherical function hλ(e) of order 2 both in H and λ. Zhang uses a generalized binomial expansion to obtain the first order terms of the expansion, and a recursion formula for the product of spherical polynomials from [8] in order to obtain the second order terms of the expansion. We should point out the work of Piotr Graczyk in [1, 2] who also investigates the central limit theorem and the expansion of the spherical functions on symmetric matrices and, in particular, on the space of positive definite matrices. In this paper, we prove the same result using a recurrence formula for the spherical functions on symmetric cones we obtained in [4]. The interest of this approach is its straightforwardness and the possibilities it opens, as a new method, for other symmetric spaces. In particular, we do not require a product formula to obtain the second order terms in the expansion. In Section 2, we recall the nature of the problem and some of the notation of [7]. In Section 3, we recall our result of [4] and explain how it can be used to compute the expansion of the spherical functions. Finally, in Section 4, we find recurrence relations which describe the coefficients of our expansion. Solving these recurrence relations is straightforward.