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Legendre Polynomials: Rodriques’ Formula and Recursion ...

Legendre Polynomials: Rodriques Formula and Recursion RelationsJackson says By manipulation of the power series solutions it is possible to obtain a compactrepresentation of the Legendre polynomials known as Rodrigues Formula . Here is a proofthat Rodrigues Formula indeed produces a solution to Legendre s differential equation. Fromthe differential equation, assuming a series solutionPn=Pajxj( = 0) we obtained therelationaj+2=j(j+ 1) n(n+ 1)(j+ 1)(j+ 2)aj[JDJ ( ), with = 0]. Withj=n 2k, this is satisfied byan 2k= ( 1)k(2n 2k)!2nk! (n k)!

Legendre Polynomials: Rodriques’ Formula and Recursion Relations Jackson says “By manipulation of the power series solutions it is possible to obtain a compact representation of the Legendre polynomials known as Rodrigues’ formula.” Here is a proof that Rodrigues’ formula indeed produces a solution to Legendre’s differential ...

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