A generalization of certain associated Bessel functions in connection with a group of shifts (Q6580576)

It is known that a Lie group depends on a finite set of parameters. The cardinality of this set is small for a low-dimensional group. In such a case, the matrix elements of representation operators, the matrix elements of bases transformations, kernels of the corresponding integral operators as well as the intertwined different realizations of representations can be expressed in terms of classical special functions. The connection between these matrix elements written in two different bases of a representation space leads to new formulas for series containing special functions.\N\NIn this paper, the authors consider a one-parameter subgroup of the group ISO\((n,1)\) and show that the kernel of an intertwined integral operator can be considered as a generalization of Macdonald and Hankel functions, in a sense that some simple cases of the kernel coincide with those variants of Bessel functions.\NUsing these connections, with different parameters, they also establish some analogues of orthogonality relations for Macdonald and Hankel functions.