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 sub(A) super( plus or minus )(a, tau ) = [int] sub(-1) super(+1) x super(a)[P sub(l') super(m') (x)] super(2)/(1 plus or minus x) super( tau ) dx, a = 0, 1, 2, 3, 4, 5, 6, tau = 1, 2, 3, I sub(B)(b, sigma ) = [int] sub(-1) super(+1) x super(b)[P sub(l') super(m') (x)] super(2)/(1 - x super(2)) super( sigma ) dx, b = 0, 2, 4, 6, 8, sigma = 1, 2, 3, and I sub(C) super( plus or minus )(c, Kappa ) = [int] sub(-1) super(+1) x super(c)[P sub(l') super(m') (x)] super(2)/[(1 - x super(2)) super( Kappa ) (1 plus or minus x)] dx, c = 0, 1, 2, 3, 4, 5, 6, 7, 8, Kappa = 1, 2. The superindices " plus or minus " in I sub(A) super( plus or minus )(a, tau ) and I sub(C) super( plus or minus ) (c, Kappa ) correspond to those of the factor (1 plus or minus 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.