## Legendre Generating Function
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.)
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 |