The LaguerreL function computes the nth Laguerre polynomial.

If the first parameter is a non-negative integer, the LaguerreL function computes the nth generalized Laguerre polynomial with parameter a evaluated at x.

If a is not specified, LaguerreL(n, x) computes the nth Laguerre polynomial which is equal to LaguerreL(n, 0, x).

The generalized Laguerre polynomials are orthogonal on the interval $\left[0,\infty \right)$ with respect to the weight function $w\left(x\right)={\ⅇ}^{-x}{x}^a$. They satisfy:

For positive integer a, the relationship for LaguerreL(n, a, x) and LaguerreL(n, x) is the following.

Some references define the generalized Laguerre polynomials differently than Maple. Denote the alternate function as altLaguerreL(n, a, x). It is defined as follows:

For general positive integer a, the relationship for Maple's LaguerreL and altLaguerreL is the following.

Laguerre polynomials satisfy the following recurrence relation:

For n not equal to a non-negative integer, the analytic extension of the Laguerre polynomial is given by:

$\mathrm{LaguerreL}\left(3\,x\right)$

$\mathrm{LaguerreL}\left(3\,x\right)$

$\mathrm{simplify}\left(\,'\mathrm{LaguerreL}'\right)$

$1-3x+\frac{3}{2}{x}^2-\frac{1}{6}{x}^3$

$\mathrm{LaguerreL}\left(3\,-\frac{1}{2}\,x\right)$

$\mathrm{LaguerreL}\left(3\,-\frac{1}{2}\,x\right)$

$\mathrm{simplify}\left(\,'\mathrm{LaguerreL}'\right)$

$\frac{5}{16}-\frac{15}{8}x+\frac{5}{4}{x}^2-\frac{1}{6}{x}^3$

$\mathrm{LaguerreL}\left(3.1\,1.2\right)$

$\mathrm{-0.7174310784}$

$\mathrm{LaguerreL}\left(2.1\,1.2\,3.4\right)$

$\mathrm{-1.498106063}$

$\mathrm{altLaguerreL}≔\left(n\,a\,x\right)\mapsto {\left(\mathrm{-1}\right)}^a\cdot n!\cdot \mathrm{LaguerreL}\left(n-a\,a\,x\right)\:$

$\mathrm{altLaguerreL}\left(3\,1\,x\right)$

$-6\mathrm{LaguerreL}\left(2\,1\,x\right)$

$\mathrm{simplify}\left(\,'\mathrm{LaguerreL}'\right)$

$-3x^2+18x-18$