Legendre Generating Function

Q: The integral

involving Legendre polynomials , is for :

Is it possible to prove that for all positive integral n?

Multiply the equation defining by , sum over , and formally interchange the order of summation and integration, to obtain

The sum can be formally evaluated using the generating function for the Legendre polynomials (see [Abramowitz and Stegun 1970, 22.9.12]):

which implies that

Setting , we can evaluate the sum by using pattern matching. (Note that the sum over in the pattern is implicit.)

Mathematica cannot compute the required definite integral. However, the indefinite integral is easily evaluated.

At the upper limit, for , this expression vanishes.

The value at the lower limit is immediate.

Hence the definite integral becomes

Since

we have shown that