Legendre Polynomials

In General [> s.a. Special Functions.]
$ Def: The solutions of the equation (think of x as cos ),

(d/dx) [(1–x²) dP_n /dx] + n(n+1) P_n(x) = 0 .

* From a generating function: The Rodrigues formula, P_n(x) = (1/2ⁿn!) dⁿ/dxⁿ (x²–1)ⁿ.
* Orthonormality: Expressed by the relationship

_–1¹ dx P_n(x) P_m(x) = 2/(2n+1) _nm .

* Examples:

P₀(x) = 1,   P₁(x) = x,   P₂(x) = (3x²–1)/2,   P₃(x) = (5x³–3x)/2.

* Properties: P_n(1) = 1, P_n(–1) = (–1)ⁿ, P_n(0) = 0 for n odd.
* Recursion relations: P'_n+1 = (2n+1) P_n + P'_n–1; (n+1) P_n+1 + n P_n–1 = (2n+1) xP_n .

Associated Legendre Functions
$ Def: The solutions of the equation

d/dx [(1–x²) dP_nm/dx] + [n(n+1) – m²/(1–x²)] P_nm(x) = 0.

* Generating function:

P_nm(x) = (–1)^m/(2ⁿn!) (1–x²)^m/2 {d^n+m/dx^n+m)(x²–1)ⁿ = (–1)^m (1–x²)^m/2 (d^m/dx^m) P_n(x) .

@ References: Saharian a0904 [summation formula].

References
@ General: in Wyld 76.
@ Modified: Yang et al mp/02 [deformed]; Durand JMP(03)mp/02 [fractional operators].
@ Related topics: Khusnutdinov JMP(03)mp [uniform asymptotic expansion].
> Online resources: see Wikipedia page.

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