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.)

`$T_j(x)T_k(x)=\frac1{2}\left(T_{j+k}(x)+T_{j-k}(x)\right)$`

which of course follows from the product to sum identities for the cosine. I will have to amend my question to asking about the closed form of the integral of the product of a Chebyshev and Legendre polynomial. $\endgroup$ – J. M. is not a mathematician Jul 27 '10 at 9:07