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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