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

Considering the kernel of an integral operator intertwining two realizations of the group of motions of the pseudo-Euclidian space, we derive two formulas for series containing Whittaker's functions or Weber's parabolic cylinder functions. We can consider this kernel as a special function. Some particular values of parameters involved in this special function are found to coincide with certain variants of Bessel functions. Using these connections, we also establish some analogues of orthogonality relations for Macdonald and Hankel functions.