The L(n, a, x) function computes the nth generalized Laguerre polynomial with parameter a evaluated at x.

In the two argument case, L(n, x) computes the nth Laguerre polynomial which is equal to L(n, 0, x).

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

For positive integer a, $L\left(n\,a\,x\right)$ is related to $L\left(n\,x\right)$ by:

Some references define the generalized Laguerre polynomials differently from Maple. Denote the alternate function as $\mathrm{altL}\left(n\,a\,x\right)$. It is defined as:

For a general positive integer a, the Maple orthopoly[L] function is related to $\mathrm{altL}$ by:

Laguerre polynomials satisfy the following recurrence relation.

$\mathrm{with}\left(\mathrm{orthopoly}\right)\:$

$L\left(3\,x\right)$

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

$L\left(15\,5\right)$

$-\frac{1998225857}{1307674368}$

$L\left(2\,1\,x\right)$

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

$L\left(11\,-\frac{1}{7}\,\frac{5}{8}\right)$

$\frac{409124992664884260142733}{27119645385619965562847232}$

$\mathrm{altL}≔\left(n\,a\,x\right)\mapsto {\left(\mathrm{-1}\right)}^a\cdot n!\cdot \mathrm{orthopoly}\left[L\right]\left(n-a\,a\,x\right)\:$

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

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