### Special Functions and Group Representations

By Eren Mehmet Kiral

March 20, 2012

#### Abstract

The Peter-Weyl theorem tells us that the matrix coefficients of irreducible finite dimensional representations of a compact group G form an orthonormal basis of functions in L^2(G). Compact or not, matrix coefficients of representations of classical lie groups turn out to be given by the so-called special functions. I will realize for example the J-Bessel function as a matrix coefficient of representations of the group of Euclidean motions of the plane, and some properties of J_n will drop out from this perspective.