Editing Classical orthogonal polynomials (section)

=== Laguerre polynomials ===
The most general Laguerre-like polynomials, after the domain has been shifted and scaled, are the Associated Laguerre polynomials (also called generalized Laguerre polynomials), denoted <math>L_n^{(\alpha)}</math>.  There is a parameter <math>\alpha</math>, which can be any real number strictly greater than &minus;1.  The parameter is put in parentheses to avoid confusion with an exponent.  The plain Laguerre polynomials are simply the <math>\alpha = 0</math> version of these:

:<math>L_n(x) = L_n^{(0)}(x).</math>

The differential equation is

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

This is '''Laguerre's equation'''.

The second form of the differential equation is

:<math>(x^{\alpha+1}\,e^{-x}\, y')' + \lambda \,x^\alpha \,e^{-x}\,y = 0.</math>

The recurrence relation is

:<math>(n+1)\,L_{n+1}^{(\alpha)}(x) = (2n+1+\alpha-x)\,L_n^{(\alpha)}(x) - (n+\alpha)\,L_{n-1}^{(\alpha)}(x).</math>

Rodrigues' formula is

:<math>L_n^{(\alpha)}(x) = \frac{x^{-\alpha}e^x}{n!} \  \frac{d^n}{dx^n}\left(x^{n+\alpha}\,e^{-x}\right).</math>

The parameter <math>\alpha</math> is closely related to the derivatives of <math>L_n^{(\alpha)}</math>:

:<math>L_n^{(\alpha+1)}(x) = - \frac{d}{dx}L_{n+1}^{(\alpha)}(x)</math>

or, more generally:

:<math>L_n^{(\alpha+m)}(x) = (-1)^m L_{n+m}^{(\alpha)[m]}(x).</math>

Laguerre's equation can be manipulated into a form that is more useful in applications:

:<math>u = x^{\frac{\alpha-1}{2}}e^{-x/2}L_n^{(\alpha)}(x)</math>

is a solution of

:<math>u'' + \frac{2}{x}\,u' + \left[\frac \lambda x - \frac{1}{4} - \frac{\alpha^2-1}{4x^2}\right]\,u = 0\text{ with } \lambda = n+\frac{\alpha+1}{2}. </math>

This can be further manipulated.  When <math>\ell = \frac{\alpha-1}{2}</math> is an integer, and <math>n \ge \ell+1</math>:

:<math>u = x^\ell e^{-x/2} L_{n-\ell-1}^{(2\ell+1)}(x)</math>

is a solution of

:<math>u'' + \frac{2}{x}\,u' + \left[\frac \lambda x - \frac{1}{4} - \frac{\ell(\ell+1)}{x^2}\right]\,u = 0\text{ with }\lambda = n.</math>

The solution is often expressed in terms of derivatives instead of associated Laguerre polynomials:

:<math>u = x^{\ell}e^{-x/2}L_{n+\ell}^{[2\ell+1]}(x).</math>

This equation arises in quantum mechanics, in the radial part of the solution of the [[Schrödinger equation]] for a one-electron atom.

Physicists often use a definition for the Laguerre polynomials that is larger, by a factor of <math>(n!)</math>, than the definition used here.

For further details, including the expressions for the first few polynomials, see [[Laguerre polynomials]].