Legendre Generating Function

Q: The integral

[Graphics:inoutgr3.gif][Graphics:inoutgr39.gif]

involving Legendre polynomials [Graphics:inoutgr40.gif], is [Graphics:inoutgr41.gif] for [Graphics:inoutgr42.gif]:

[Graphics:inoutgr3.gif][Graphics:inoutgr43.gif]
[Graphics:inoutgr3.gif][Graphics:inoutgr44.gif]

Is it possible to prove that [Graphics:inoutgr45.gif] for all positive integral n?

Multiply the equation defining [Graphics:inoutgr46.gif] by [Graphics:inoutgr47.gif], sum over [Graphics:inoutgr48.gif], and formally interchange the order of summation and integration, to obtain

[Graphics:inoutgr3.gif][Graphics:inoutgr49.gif]

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

[Graphics:inoutgr3.gif][Graphics:inoutgr50.gif]

which implies that

[Graphics:inoutgr3.gif][Graphics:inoutgr51.gif]

Setting [Graphics:inoutgr52.gif], we can evaluate the sum by using pattern matching. (Note that the sum over [Graphics:inoutgr53.gif] in the pattern is implicit.)

[Graphics:inoutgr3.gif][Graphics:inoutgr54.gif]
[Graphics:inoutgr3.gif][Graphics:inoutgr55.gif]

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

[Graphics:inoutgr3.gif][Graphics:inoutgr56.gif]
[Graphics:inoutgr3.gif][Graphics:inoutgr57.gif]

At the upper limit, for [Graphics:inoutgr58.gif], this expression vanishes.

[Graphics:inoutgr3.gif][Graphics:inoutgr59.gif]
[Graphics:inoutgr3.gif][Graphics:inoutgr60.gif]
[Graphics:inoutgr3.gif][Graphics:inoutgr61.gif]
[Graphics:inoutgr3.gif][Graphics:inoutgr62.gif]
[Graphics:inoutgr3.gif][Graphics:inoutgr63.gif]
[Graphics:inoutgr3.gif][Graphics:inoutgr64.gif]

The value at the lower limit is immediate.

[Graphics:inoutgr3.gif][Graphics:inoutgr65.gif]
[Graphics:inoutgr3.gif][Graphics:inoutgr66.gif]

Hence the definite integral becomes

[Graphics:inoutgr3.gif][Graphics:inoutgr67.gif]
[Graphics:inoutgr3.gif][Graphics:inoutgr68.gif]

Since

[Graphics:inoutgr3.gif][Graphics:inoutgr69.gif]
[Graphics:inoutgr3.gif][Graphics:inoutgr70.gif]

we have shown that

[Graphics:inoutgr3.gif][Graphics:inoutgr71.gif]