On some integral formulae containing Bessel functions

Abstract

So called Sounine's integral formula can be interpreted as follows [1]. Let Gn be n-dimensional Euclidean motion group and Tg be an irreducible unitary representaion of class 1. The group Gn contains the subgroup which is isomorphic to Gn–1. Being restricted to Gn–1, Tg is decomposed into irreducible representations of Gn–1. Using this decomposition, we can express the zonal spherical function of Gn by that of Gn–1. This expression is nothing but Sonnine's first integral formula. Sonnine's second integral formula can be proved in the same way.

On the other hand, the motion group of n-dimensional Lovachvsky space G (the Lorentz group) contains, as a subgroup, Gn–1. As in the previous case, restricting the irreducible representation of G to Gn–1 and decomposing it into irreducible factors, we can obtain an integral formula (6) involving Legendre and Bessel functions. In this sense, we may call the formula (6) an analogue to Sonnine's formula.

If we consider imaginary Lovachevsky space, we obtain analogous formula (7), (8).