Is there a closed form for the quadruple inner product of Legendre Polynomials such as:
\begin{align}
\int_{-1}^{1}P_k(x)P_l(x)P_m(x)P_n(x)dx
\end{align}
I am aware of solutions for the triple inner product for Legendre Polynomials.
closed-formlegendre polynomialsorthogonal-polynomials
Is there a closed form for the quadruple inner product of Legendre Polynomials such as:
\begin{align}
\int_{-1}^{1}P_k(x)P_l(x)P_m(x)P_n(x)dx
\end{align}
I am aware of solutions for the triple inner product for Legendre Polynomials.
Best Answer
Take a look at "J. Miller, Formulas for Integrals of Products of Associated Legendre or Laguerre Functions, Mathematics of Computation, Vol. 17, No. 81 (Jan., 1963), pp. 84-87" and take $m=0$ and $r=4$.