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

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 $({\mathrm{SO}}_0(n,1),\mathrm{SO}(n))$.