On Integrals Involving Universal Associated Legendre Polynomials and Powers of the Factor (1 − x2) and Their Byproducts**Supported by the National Natural Science Foundation of China under Grant No. 11275165 and Partially by 20160978-SIP-IPN, Mexico

Abstract

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 Pl'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.