Some results involving generalized associated Legendre functions

Abstract

In this paper, a new generalization of the associated Legendre functions of the first and the second kinds is introduced using the r-generalized Gauss hypergeometric function. The basic properties of these functions, in particular, some recurrence relations and differential and integral representations, are given. The Whipple formulae are established. A new generalization of the classical Mehler-Fock integral transform is constructed, and the inversion formula is proved. Some new integrals involving the functions τ,β r P μ ν (t) and τ,β r P m,n k (z) are evaluated.