Legendre Polynomials

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

\[\def\dd{{\rm d}}{\dd\over\dd x}\,\Big[(1-x^2)\,{\dd P_n\over\dd x}\Big] + n(n+1)\,P_n(x) = 0\;.\]

* 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) δ_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

\[{\dd\over\dd x}\,\Big[(1-x^2)\,{\dd P_{nm}\over\dd x}\Big] + \Big[n(n+1)-{m^2\over(1-x^2)}\Big]\,P_{nm}(x) = 0\;.\]

* 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 JPA(09)-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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