Generating Function for Legendre Polynomials

Theorem

Let $\map {P_n} x$ denote the $n$th Legendre polynomial.

Then the generating function for $P_n$ is:

$\ds \frac 1 {\sqrt {1 - 2 x t + t^2} } = \sum_{k \mathop = 0}^\infty \map {P_k} x t^k$

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): generating function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): generating function