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~m＇l_′（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-x~2）~（-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.