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.

• 22E45 - Representations of Lie and linear algebraic groups over real fields: analytic methods
• 33B15 - Gamma, beta and polygamma functions
• 33C05 - Classical hypergeometric functions, \({}_2F_1\)
• 33C10 - Bessel and Airy functions, cylinder functions, \({}_0F_1\)
• 33C80 - Connections of hypergeometric functions with groups and algebras, and related topics
• 33D52 - Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)

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