Legendre Polynomials Triple Product

I have to solve the following integral:

\begin{align} \int_{-1}^{1} \left(x^2 -1\right)^3 P_k(x)\,P_l(x)\, P_m(x) \;dx \end{align} where $P_{k,l,m}$ are Legendre Polynomials

The triple product \begin{align} \int_{-1}^{1} P_k(x)\,P_l(x)\, P_m(x) \;dx = 2 \begin{pmatrix} k & l & m \\ 0 & 0 & 0 \end{pmatrix}^2 \end{align} using the special case of $3j$ symbol form \begin{align} \begin{pmatrix} k & l & m \\ 0 & 0 & 0 \end{pmatrix} &= (-1)^s \sqrt{(2s-2k)! (2s-2l)! (2s-2m)! \over (2s+1)!} {s! \over (s-k)! (s-l)! (s-m)!} \\ & \mbox{for $2s=k+l+m$ even} \\[3pt] \begin{pmatrix} k & l & m \\ 0 & 0 & 0 \end{pmatrix} &= 0 \quad\mbox{for $2s=k+l+m$ odd} \\ \end{align}

I'm sure you should be able to solve this by doing integration by parts but can't seem to get it to work. Any tips?

So using the answer below I think you get the following for step 1 of 3

\begin{multline} \int_{-1}^{1}(x^2-1)^3 P_k P_l P_m = \overbrace{(x^2-1)^3\frac{(P_{k+1} - P_{k-1})}{2k+1} P_l P_m \Big]_{-1}^1}^\text{ = 0}\\ -\int_{-1}^{1} \frac{(P_{k+1} - P_{k-1})}{2k+1}(x^2-1)^2\Big( 6xP_l P_m \\ + (1+l) P_m(P_{l+1} - P_{l-1}) + (1+m) P_l(P_{m+1} - P_{m-1}) \Big) \; dx\\ \end{multline}

Not sure if the formula for integration works as I think the $6xP_lP_m$ term might cause problems?

Solution 1:

I think the best way to approach this is as follows, note that \begin{align} (x^2 -1 ) = \frac{P_2 - 2}{3} \end{align} You can then use the following definition \begin{align} P_kP_l = \sum_{m=|k-l|}^{k+l} \begin{pmatrix} k & l & m \\ 0 & 0 & 0 \end{pmatrix}^2 (2m+1)P_m \end{align} This allows the integral to be written as follows \begin{align} \int_{-1}^{1} (x^2-1)^3P_iP_jP_k \; dx &= \int_{-1}^{1} \frac{1}{9}\left(P_2^3 + . . .-8 \right) P_i P_j P_k \; dx \end{align} The most difficult term to deal with is the $ P_2^3 P_i P_j P_k$ \begin{align} P_2^3 P_i P_j P_k &= \sum_{m=0}^{4} \begin{pmatrix} 2 & 2 & m \\ 0 & 0 & 0 \end{pmatrix}^2 (2m+1)P_m P_2 P_i P_j P_k \\ &= \sum_{m=0}^{4} \begin{pmatrix} 2 & 2 & m \\ 0 & 0 & 0 \end{pmatrix}^2 (2m+1) \sum_{n=|m-2|}^{m+2} \begin{pmatrix} 2 & m & n \\ 0 & 0 & 0 \end{pmatrix}^2 (2n+1)P_n P_i P_j P_k \\ &= \sum_{m=0}^{4} \begin{pmatrix} 2 & 2 & m \\ 0 & 0 & 0 \end{pmatrix}^2 (2m+1) \sum_{n=|m-2|}^{m+2} \begin{pmatrix} 2 & m & n \\ 0 & 0 & 0 \end{pmatrix}^2 (2n+1) \sum_{l=|n-i|}^{n+i} \begin{pmatrix} n & i & l \\ 0 & 0 & 0 \end{pmatrix}^2 (2l+1) P_l P_j P_k \end{align} Which can then make use of the usual triple integral. All other terms can be solved for in a similar manner.

Solution 2:

You could integrate one of the $P_k(x)$ and take the derivative of the rest. The power of $(1-x^2)$ gets reduce by the fact that $$\partial_x P_l(x) = \frac{(1+l) [ P_{l+1}(x)-x P_l(x) ]}{x^2-1}.$$ You have to apply partial integration a few times and you will generate a bunch of Legendre triple product.