Editing Classical orthogonal polynomials (section)

=== Legendre polynomials ===
The differential equation is

:<math>(1-x^2)\,y'' - 2x\,y' + \lambda \,y = 0\qquad \text{with}\qquad\lambda = n(n+1).</math>

This is '''Legendre's equation'''.

The second form of the differential equation is:

:<math>\frac{d}{dx}[(1-x^2)\,y'] + \lambda\,y = 0.</math>

The [[recurrence relation]] is

:<math>(n+1)\,P_{n+1}(x) = (2n+1)x\,P_n(x) - n\,P_{n-1}(x).</math>

A mixed recurrence is

:<math>P_{n+1}^{[r+1]}(x) = P_{n-1}^{[r+1]}(x) + (2n+1)\,P_n^{[r]}(x).</math>

Rodrigues' formula is

:<math>P_n(x) = \,\frac{1}{2^n n!} \  \frac{d^n}{dx^n}\left([x^2-1]^n\right).</math>

For further details, see [[Legendre polynomials]].

==== Associated Legendre polynomials ====
The [[Associated Legendre polynomials]], denoted
<math>P_\ell^{(m)}(x)</math> where <math>\ell</math> and <math>m</math> are integers with <math>0 \leqslant m  \leqslant \ell</math>, are defined as

:<math>P_\ell^{(m)}(x) = (-1)^m\,(1-x^2)^{m/2}\ P_\ell^{[m]}(x).</math>

The ''m'' in parentheses (to avoid confusion with an exponent) is a parameter.  The ''m'' in brackets denotes the ''m''-th derivative of the Legendre polynomial.

These "polynomials" are misnamed—they are not polynomials when ''m'' is odd.

They have a recurrence relation:

:<math>(\ell+1-m)\,P_{\ell+1}^{(m)}(x) = (2\ell+1)x\,P_\ell^{(m)}(x) - (\ell+m)\,P_{\ell-1}^{(m)}(x).</math>

For fixed ''m'', the sequence <math>P_m^{(m)}, P_{m+1}^{(m)}, P_{m+2}^{(m)}, \dots</math> are orthogonal over [&minus;1,&nbsp;1], with weight&nbsp;1.

For given ''m'', <math>P_\ell^{(m)}(x)</math> are the solutions of

:<math>(1-x^2)\,y'' -2xy' + \left[\lambda - \frac{m^2}{1-x^2}\right]\,y = 0\qquad \text{ with }\qquad\lambda = \ell(\ell+1).</math>