Do the following integrals have a closed-form solution for any integer value of $m,l,k$ and $n$?

$\int^{\pi}_{0} P^{m}_{l}\left(\cos\theta\right)P^{n}_{k}\left(\cos\theta\right)\cot\theta d\theta$

$\int^{\pi}_{0} P^{m}_{l}\left(\cos\theta\right)P^{n}_{k}\left(\cos\theta\right)\frac{1}{\sin\theta} d\theta$

  • 2
    $\begingroup$ What do you mean by "non-singular solution"? The integrals are singular, but if they converge, they just have a solution. Maybe you mean "closed form"? $\endgroup$ – Wolfgang Aug 9 '16 at 19:26
  • $\begingroup$ Surely you are integrating from on $[0,\pi]$ rather than $[-1,1]$? In any case I'd change variables to $x=\cos\theta$. $\endgroup$ – Lior Silberman Aug 9 '16 at 20:45

The first integral can be evaluated by using formulas (13) and (8) from http://www.sciencedirect.com/science/article/pii/S0893965998001803 (A generalized formula for the integral of three associated Legendre polynomials, by H.A. Mavromatis and R.S. Alassar). Some special cases of the second integral are given in http://www.jstor.org/stable/2005832 ( Some Integrals Involving Associated Legendre Functions, by S. N. Samaddar). Maybe the following article http://iopscience.iop.org/article/10.1088/0305-4470/19/13/016 (Evaluation of integrals involving powers of (1-x2) and two associated Legendre functions or Gegenbauer polynomials, by M.A. Rashid) will be also of some use.

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