 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].