Closed form for an orthogonal polynomial integral? The following integral came up in one of my applications:
$\int_{-1}^1P_n(x)T_j(x)T_k(x)\mathrm{d}x$
where $P_n(x)$ is a Legendre polynomial, $T_k(x)$ is a Chebyshev polynomial, and $j$, $k$, and $n$ are nonnegative integers.
I want to ask if there might be a closed-form representation for this integral. I have a feeling it will involve gamma functions and Pochhammer symbols, but I seem to be unable to figure out how to proceed.
Alternatively, since I am aware that Legendre polynomials can be expressed as a linear combination of Chebyshev polynomials, it might be easier to instead simplify the integral
$\int_{-1}^1T_n(x)T_j(x)T_k(x)\mathrm{d}x$
or in trigonometric form
$\int_{0}^{\pi}\cos(n\theta)\cos(j\theta)\cos(k\theta)\sin(\theta)\mathrm{d}\theta$
but I do not know of any closed form for this either.
I have already tried looking in Abramowitz and Stegun, the DLMF, Gradshteyn and Ryzhik, and the Wolfram functions site to no avail.
(edit:
I had neglected to exploit the identity
$T_j(x)T_k(x)=\frac1{2}\left(T_{j+k}(x)+T_{j-k}(x)\right)$
when I first formulated my question. I now amend my question to asking for a closed form for
$\int_{-1}^1P_n(x)T_j(x)\mathrm{d}x$
of which the only fact I know about it is that it is 0 if $j<n$ by virtue of the orthogonality of the Legendre polynomial.)
 A: It turns out the identities I needed for resolving
$\int_{-1}^1P_n(x)T_j(x)\mathrm{d}x$
into a closed form was well-hidden in Abramowitz and Stegun and Gradshteyn and Ryzhik.
As I had mentioned in the edit to my original question, the integral is 0 if $j<n$ by virtue of the orthogonality property of the Legendre polynomials.
I now considered the following integral:
$\int_{-1}^1P_n(x)T_{n+k}(x)\mathrm{d}x\quad k\geq0$
To dispose of an elementary case first, I noted that $P_n(x)T_{n+k}(x)$ is an odd function iff $k$ is odd and even iff $k$ is even; the integral is therefore 0 for odd $k$.
The even $k$ case I had solved by making use of two identities: this series representation for $T_{n}(x)$ (also in Abramowitz and Stegun as 22.3.6), and an integral I derived from a more general form in Gradshteyn and Ryzhik:
$\int_0^1x^{n+2\rho}P_n(x)\mathrm{d}x=\frac{\left(2\rho+1\right)_n}{2^{n+1}\left(\rho+\frac1{2}\right)_{n+1}}$
where $\left(a\right)_n$ is the Pochhammer symbol. (The identity actually listed in G&R was an integral for a Gegenbauer (ultraspherical) polynomial, of which the Legendre polynomial is a special case.)
I only needed to retain terms in the series greater than or equal to $n$, again due to orthogonality of the Legendre polynomial. Applying the integral formula to each term (with an additional factor of 2 because the integrand is even), and feeding the resulting sum to Mathematica netted the following closed form:
$\int_{-1}^1P_n(x)T_{n+2k}(x)\mathrm{d}x=-\frac1{4}\frac{\left(n+2k\right)\Gamma\left(n+k\right)\Gamma\left(k-\frac1{2}\right)}{\Gamma\left(k+1\right)\Gamma\left(n+k+\frac{3}{2}\right)}$
(The original result returned by Mathematica 5.2 had nasty cosecant factors, which I disposed of using the reflection formula for the gamma function).
This can then be applied to the original integral with the three polynomials by exploiting the product-sum identity for the Chebyshev polynomial.
