Some Connections between the Spherical and Parabolic Bases on the Cone Expressed in terms of the Macdonald Function

Abstract

Computing the matrix elements of the linear operator, which transforms the spherical basis of $SO(3,1)$-representation space into the hyperbolic basis, very recently, Shilin and Choi (2013) presented an integral formula involving the product of two Legendre functions of the first kind expressed in terms of ${\hspace{0.17em}}_4{F}_3$-hypergeometric function and, using the general Mehler-Fock transform, another integral formula for the Legendre function of the first kind. In the sequel, we investigate the pairwise connections between the spherical, hyperbolic, and parabolic bases. Using the above connections, we give an interesting series involving the Gauss hypergeometric functions expressed in terms of the Macdonald function.