Integration of Legendre Polynomial I need to evaluate the following integral
\begin{equation} \int_{-1}^1 \frac{d^4P_l(x)}{dx^4}P_n(x)dx\end{equation}. Of course the answer I need is in terms of $l$ and $n$. Does anyone have any idea how to proceed?

Let me see if I can give a solution. The orthogonality relation for the Gegenbauer polynomials is
$$
\int_{-1}^{1} P_m(x) P_n(x) dx = \frac{2}{2n+1} \delta_{m,n}.
$$
The forward shift operator for the Jacobi polynomials, e.g. http://homepage.tudelft.nl/11r49/askey/ch1/par8/par8.html, is given by
$$
\frac{d}{dx} P_m^{(\alpha,\beta)}(x) = \frac{n+\alpha+\beta+1}{2} P_{m-1}^{(\alpha+1,\beta+1)}(x).
$$
Note that the Gegenbauer polynomials are Jacobi polynomial where $\alpha = \beta = 0$.
The connection coefficients for the Jacobi polynomials are
$$
P_n^{(\gamma,\delta)}(x) = \sum_{k=0}^n c_{n,k}(\gamma,\delta;\alpha,\beta) P_k^{(\alpha,\beta)}(x),
$$
where
\begin{equation*}
\begin{split}
c_{n,k}(\gamma,\delta;\alpha,\beta) &= \frac{
  (\gamma+k+1)_{n-k} (n+\gamma+\delta+1)_k
}{
  (n-k)! \Gamma(\alpha+\beta+2k+1)
} \Gamma(\alpha+\beta+k+1) \\
&\times {}_3 F_2(-n+k, n+k+\gamma+\delta+1, \alpha+k+1; \gamma+k+1, \alpha+\beta+2k+2; 1).
\end{split}
\end{equation*}
Hence
\begin{equation*}
\begin{split}
\frac{d^4}{dx^4} P_{\ell}^{(0,0)}
  &= \frac{(\ell-2)_4}{2^4} P_{\ell-4}^{(4,4)}(x)
  = \frac{(\ell-2)_4}{2^4} \sum_{k=0}^{\ell-4} c_{\ell-4,k}(4,4;0,0) P_k^{(0,0)}(x).
\end{split}
\end{equation*}
By the orthogonality relation we have
\begin{equation*}
\int_{-1}^{1} \frac{d^4}{dx^4} P_{\ell}(x) P_n(x) dx = \frac{(\ell-2)_4}{2^3 (2n+1)} c_{\ell-4,n}(4,4;0,0).
\end{equation*}
Note that the connection coefficient can be found in Ismail, Classical and Quantum Orthogonal polynomials and that the connection coefficients probably can be simplified a lot. I hope I don't divide by zero somewhere so that you have to intrepid it formally. So the last part is homework. ;)
