Closed Form Integral

Q: How can I compute a closed expression for

where is a Legendre polynomial?

A: Orthogonality of the Legendre polynomials requires that

One can write as a Gaussian hypergeometric function (functions.wolfram.com/Polynomials/LegendreP/26/01/01/0001),

Specializing equation (2.5) of Fields and Wimp [1], to the case where and , one has

Put , , , , , into this formula.

The term causes the summand to vanish if , making the sum finite. We simplify the summand as follows.

It is a good idea to verify that this expression is correct, at least for small .

Finally, using the parity property, , we obtain the expansion coefficients .

We check this result for .