Mehler-Heine formula: a generalization in the context of spherical functions

TL;DR

This paper generalizes the Mehler-Heine formula by deriving spherical functions of certain Gelfand pairs as limits of other spherical functions using group contraction techniques.

Contribution

It introduces a novel approach to obtain spherical functions of specific Gelfand pairs as limits of functions from related pairs through group contraction.

Findings

Spherical functions of (M(n), SO(n)) are derived as limits of those of (SO(n+1), SO(n)).

Spherical functions of (M(n), SO(n)) are also obtained as limits of those of (SO_0(n,1), SO(n)).

The method employs group contraction to connect different Gelfand pairs.

Abstract

In this article, using the notion of group contraction, we obtain the spherical functions of the strong Gelfand pair $(\mathrm{M}(n),\mathrm{SO}(n))$ as an appropriate limit of spherical functions of the strong Gelfand pair $(\mathrm{SO}(n+1),\mathrm{SO}(n))$ and also of the strong Gelfand pair $(\mathrm{SO}_0(n,1),\mathrm{SO}(n))$.