Zonal and Associated Functions on $SO_{0}(p,q)$ Groups

TL;DR

This paper derives explicit formulas for spherical functions on $SO(p,q)$ groups using hypergeometric series, and explores their applications to de Sitter and conformal groups, including a theorem on distribution derivatives under infinite-dimensional Lie groups.

Contribution

It provides new explicit expressions for zonal and associated spherical functions on $SO(p,q)$ groups and introduces a theorem on distribution derivatives related to infinite-dimensional Lie groups.

Findings

Explicit formulas for spherical functions using hypergeometric series.

Application to de Sitter and conformal invariance groups.

A new theorem on distribution derivatives on smooth surfaces.

Abstract

Explicit expressions for associated spherical functions of $SO(p,q)$ matrix groups are obtained using a generalized hypergeometric series of two variables. In this paper, we present explicit expressions for zonal functions of de Sitter groups and the group of conformal invariance. Moreover, we present a theorem on the transformation of derivative of distributions, concentrated on smooth surfaces, with respect to infinite-dimensional Lie group $C^{\infty}\left(\mathbb{R^{\mathit{n}}},GL(n)\right)$.