# Berezin-Karpelevich formula for chi-spherical functions on complex Grassmannians

### From NZJM

**New Zealand Journal of Mathematics**

**Vol.** 50, (2020), **Pages** 29-48

**Mahmoud Al-Hashami**

Department of Mathematics

Sultan Qaboos University

Alkhoud

Sultanate of Oman

mailto:s100111@student.squ.edu.om

**Abstract** In [5], Berezin and Karpelevich gave, without a proof, an explicit formula for spherical functions on complex Grassmannian manifolds. A first attempt to give a proof of Berezin-Karpelevich formula was taken, in [16], by Takahashi. His proof contained a gap, which was fixed later, in [10], by Hoogenboom. The aim of this paper is to generalize Berezin-Karpelevich formula to the case of χ-spherical functions on complex Grassmannian manifolds .

**Keywords** Spherical Functions, Hermitian Symmetric Spaces, Jacobi Function, Laplace-Beltrami Operator.

**Classification** (MSC2000) 43A85, 28C10, 43A77, 43A90, 53C35.