Generalized Bessel functions and the representation theory of U(2) σ C2×2

Abstract

We construct the matrix elements of both finite transformations and infinitesimal generators in irreducible representations of the motion group U(2) σ C2×2 with the aid of the contraction limit of the analogous structures of U(4). The matrix elements of finite transformations are found to have a structure similar to that of the classical Bessel function in that they contain two inverse gamma matrices which couple Wigner D functions. An integral representation is established and related to the matrix-valued Bessel functions of Gross and Kunze. By means of the representation property of the matrix elements we obtain a new sum rule for classical Bessel functions and an analog of the binomial theorem for the sum of two 2×2 matrices which involves the U(2) gamma matrix instead of the classical gamma function.