Universal Associated Legendre Polynomials and Some Useful Definite Integrals

We first introduce the universal associated Legendre polynomials, which are occurred in studying the non-central fields such as the single ring-shaped potential and then present definite integrals I_A ^±(a, τ) = ∫_-1 ⁺¹ x^a[P_l' ^m' (x)]²/(1 ± x)^τ dx, a = 0, 1, 2, 3, 4, 5, 6, τ = 1, 2, 3, I_B(b, σ) = ∫_-1 ⁺¹ x^b[P_l' ^m' (x)]²/(1 - x²)^σ dx, b = 0, 2, 4, 6, 8, σ = 1, 2, 3, and I_C ^±(c, κ) = ∫_-1 ⁺¹ x^c[P_l' ^m' (x)]²/[(1 - x²)^κ (1 ± x)] dx, c = 0, 1, 2, 3, 4, 5, 6, 7, 8, κ = 1, 2. The superindices "±" in I_A ^±(a, τ) and I_C ^± (c, κ) correspond to those of the factor (1 ± x) involved in weight functions. The formulas obtained in this work and also those for integer quantum numbers l' and m' are very useful and unavailable in classic handbooks.