Ring-Shaped Potential and a Class of Relevant Integrals Involved Universal Associated Legendre Polynomials with Complicated Arguments

We find that the solution of the polar angular differential equation can be written as the universal associated Legendre polynomials. Its generating function is applied to obtain an analytical result for a class of interesting integrals involving complicated argument, that is, ∫ - 1 1 P l ′ m ′ x t - 1 / 1 + t 2 - 2 x t P k ′ m ′ (x) / (1 + t 2 - 2 t x) (l ′ + 1) / 2 d x, where t ∈ (0,1). The present method can in principle be generalizable to the integrals involving other special functions. As an illustration we also study a typical Bessel integral with a complicated argument ∫ 0 ∞ J n (α x 2 + z 2) / (x 2 + z 2) n x 2 m + 1 d x.