On integrals involving universal associated legendre polynomials and powers of the factor (1 - X ²) and their byproducts

The associated Legendre polynomials play an important role in the central fields, but in the case of the non-central field we have to introduce the universal associated Legendre polynomials P_l' ^m'(x) when studying the modified Pschl-Teller potential and the single ring-shaped potential. We present the evaluations of the integrals involving the universal associated Legendre polynomials and the factor (1-x²)^-p-1 as well as some important byproducts of this integral which are useful in deriving the matrix elements in spin-orbit interaction. The calculations are obtained systematically using some properties of the generalized hypergeometric series.