Expansion of a class of functions into an integral involving associated Legendre functions (Q1326618)

The main result involves an inversion formula for the index transformation \[ F(\sigma) = \int^{x_ 2}_{x_ 1} {f(x) \over 1- x^ 2} M(x,x_ 1; i \sigma) dx, \quad -1<x_ 1<x_ 2<1, \] where the kernel \[ M(x,y;i \sigma)=P^{i \sigma}_{-1/2 + i \tau} (x) P^{i \sigma}_{-1/2 + i \tau} (-y)- P^{i \sigma}_{-1/2 + \tau} (-x) P^{i \sigma}_{-1/2 + i \tau} (y) \] is expressed in terms of Legendre functions.