How can I evaluate
$$ \int\_{-1}^1 P\_n(x)P\_l(x)x^k dx $$
when $k$ is even?


Or what might be a source where I could find integrals like this? 



Legendre Polynomial Integral
The integral $$\int\limits\_0^1 x^k P\_m(x)P\_n(x)dx$$ is evaluated in terms of the hypergeometric function $\_3F\_2$ in <http://link.springer.com/article/10.1007/BF01650571> (Some integrals containing products of legendre polynomials, by L. Carlitz). The evaluation is based on
$$P\_m(x)P\_n(x)=\sum\limits\_{r=0}^{\mathrm{min}(m,n)}\frac{A\_rA\_{m-r}A\_{n-r}}{A\_{m+n-r}}\,\frac{2m+2n-4r+1}{2m+2n-2r+1}P\_{m+n-2r}(x),$$
with $$A\_r=\frac{(2r-1)!!}{2^rr!},$$
and $$\int\limits\_0^1 x^k P\_n(x)dx=\frac{k(k-1)\cdots(k-n+2)}{(k+n+1)(k+n-1)\cdots(k-n+3)}.$$
The following formulas
$$\int\limits\_{-1}^1 x P\_m(x)P\_n(x)dx=\left\{\begin{array}{cc}\frac{2(m+1)}{(2m+1)(2m+3)}, & n=m+1, \\ \frac{2m}{(2m-1)(2m+1)}, & n=m-1,\end{array}\right . $$ and
$$\int\limits\_{-1}^1 x^2 P\_m(x)P\_n(x)dx=\left\{\begin{array}{cc}\frac{2(m+1)(m+2)}{(2m+1)(2m+3)(2m+5)}, & n=m+2, \\ 
\frac{2(2m^2+2m-1)}{(2m-1)(2m+1)(2m+3)}, & n=m, \\
\frac{2m(m-1)}{(2m-3)(2m-1)(2m+1)}, & n=m-2,\end{array}\right . $$ are given as an exercise 12.9.2 (p. 806) in the book Mathematical Methods for Physicists, 6th Edition, by G.B Arfken and H.J. Weber: [https://www.amazon.com/Mathematical-Methods-Physicists-George-Arfken/dp/0120598760](http://rads.stackoverflow.com/amzn/click/0120598760)


