SOME FORMULAS FOR BESSEL AND HYPER-BESSEL FUNCTIONS RELATED TO THE PROPER LORENTZ GROUP

Abstract

In a representation space of the proper Lorentz group, we consider the so-called spherical and two parabolic bases and compute the matrix elements of restriction of the representation to matrix diag(1, 1, 1, −1) with respect to one of the above parabolic basis in the following three particular cases: matrix elements belong to ‘zero row’; lie on the ‘main diagonal’; lie on the ‘anti-diagonal’. Taking the relations between above bases, we give a group theoretical treatment of one known formula and derive two new formulas for series involving modified hyper-Bessel functions of the first kind, which converge to products of (usual) cylinder functions. Some results here are pointed out to be able to be rewritten in terms of Bessel-Clifford functions.