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, Pn(x) = (1/2nn!) dn/dxn (x2–1)n.
* Orthonormality: Expressed by the relationship

–11 dx Pn(x) Pm(x) = 2/(2n+1) δnm .

* Examples:

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

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

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:

Pnm(x) = (–1)m/(2nn!) (1–x2)m/2 {dn+m/dxn+m)(x2–1)n = (–1)m (1–x2)m/2 (dm/dxm) Pn(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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