LaguerreL - Maple Help

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LaguerreL

Laguerre function

 Calling Sequence LaguerreL(n, a, x)

Parameters

 n - algebraic expression a - (optional) nonrational algebraic expression or rational number x - algebraic expression

Description

 • 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:

${\int }_{0}^{\mathrm{\infty }}w\left(t\right)\mathrm{LaguerreL}\left(m,a,t\right)\mathrm{LaguerreL}\left(n,a,t\right)ⅆt=\left\{\begin{array}{cc}0& n\ne m\\ \frac{\mathrm{\Gamma }\left(n+a+1\right)}{n!}& n=m\end{array}$

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

$\mathrm{LaguerreL}\left(n,a,x\right)={\left(-1\right)}^{a}\left(\frac{{\partial }^{a}}{\partial {x}^{a}}\mathrm{LaguerreL}\left(n+a,x\right)\right)$

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

$\mathrm{altLaguerreL}\left(n,a,x\right)=\frac{{\partial }^{a}}{\partial {x}^{a}}\mathrm{altLaguerreL}\left(n,x\right)$

$\mathrm{altLaguerreL}\left(n,x\right)=n!\mathrm{LaguerreL}\left(n,x\right)$

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

$\mathrm{altLaguerreL}\left(n,a,x\right)={\left(-1\right)}^{a}n!\mathrm{LaguerreL}\left(n-a,a,x\right)$

 • Laguerre polynomials satisfy the following recurrence relation:

$\mathrm{LaguerreL}\left(0,a,x\right)=1,$

$\mathrm{LaguerreL}\left(1,a,x\right)=-x+1+a,$

$\mathrm{LaguerreL}\left(n,a,x\right)=\frac{\left(2n+a-1-x\right)}{n}\mathrm{LaguerreL}\left(n-1,a,x\right)-\frac{\left(n+a-1\right)}{n}\mathrm{LaguerreL}\left(n-2,a,x\right),\mathrm{for}n>1.$

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

$\mathrm{LaguerreL}\left(n,a,x\right)=\mathrm{binomial}\left(n+a,n\right)\mathrm{KummerM}\left(-n,a+1,x\right)$

Examples

 > $\mathrm{LaguerreL}\left(3,x\right)$
 ${\mathrm{LaguerreL}}{}\left({3}{,}{x}\right)$ (1)
 > $\mathrm{simplify}\left(,'\mathrm{LaguerreL}'\right)$
 ${1}{-}{3}{}{x}{+}\frac{{3}}{{2}}{}{{x}}^{{2}}{-}\frac{{1}}{{6}}{}{{x}}^{{3}}$ (2)
 > $\mathrm{LaguerreL}\left(3,-\frac{1}{2},x\right)$
 ${\mathrm{LaguerreL}}{}\left({3}{,}{-}\frac{{1}}{{2}}{,}{x}\right)$ (3)
 > $\mathrm{simplify}\left(,'\mathrm{LaguerreL}'\right)$
 $\frac{{5}}{{16}}{-}\frac{{15}}{{8}}{}{x}{+}\frac{{5}}{{4}}{}{{x}}^{{2}}{-}\frac{{1}}{{6}}{}{{x}}^{{3}}$ (4)
 > $\mathrm{LaguerreL}\left(3.1,1.2\right)$
 ${-}{0.7174310784}$ (5)
 > $\mathrm{LaguerreL}\left(2.1,1.2,3.4\right)$
 ${-}{1.498106063}$ (6)

Using the alternate definition for the Laguerre polynomials:

 > $\mathrm{altLaguerreL}≔\left(n,a,x\right)→{\left(-1\right)}^{a}n!\mathrm{LaguerreL}\left(n-a,a,x\right):$
 > $\mathrm{altLaguerreL}\left(3,1,x\right)$
 ${-}{6}{}{\mathrm{LaguerreL}}{}\left({2}{,}{1}{,}{x}\right)$ (7)
 > $\mathrm{simplify}\left(,'\mathrm{LaguerreL}'\right)$
 ${-}{3}{}{{x}}^{{2}}{+}{18}{}{x}{-}{18}$ (8)