Some results involving generalized associated Legendre functions

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 ν μ (t) and are evaluated.