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orthopoly

 L
 Laguerre polynomial

 Calling Sequence L(n, a, x) L(n, x)

Parameters

 n - non-negative integer a - rational number greater than -1 or nonrational algebraic expression x - algebraic expression

Description

 • 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:

${\int }_{0}^{\mathrm{\infty }}w\left(t\right)L\left(m,a,t\right)L\left(n,a,t\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt=\left\{\begin{array}{cc}0& n\ne m\\ \frac{\mathrm{\Gamma }\left(a+n+1\right)}{n!}& n=m\end{array}\right\$

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

$L\left(n,a,x\right)={\left(-1\right)}^{a}\frac{{ⅆ}^{a}}{ⅆ{x}^{a}}L\left(n+a,x\right)$

 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:

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

$\mathrm{altL}\left(n,x\right)=n!L\left(n,x\right)$

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

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

 • Laguerre polynomials satisfy the following recurrence relation.

$L\left(0,a,x\right)=1,$

$L\left(1,a,x\right)=-x+1+a,$

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

Examples

 > $\mathrm{with}\left(\mathrm{orthopoly}\right):$
 > $L\left(3,x\right)$
 ${1}{-}{3}{}{x}{+}\frac{{3}}{{2}}{}{{x}}^{{2}}{-}\frac{{1}}{{6}}{}{{x}}^{{3}}$ (1)
 > $L\left(15,5\right)$
 ${-}\frac{{1998225857}}{{1307674368}}$ (2)
 > $L\left(2,1,x\right)$
 ${3}{-}{3}{}{x}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}$ (3)
 > $L\left(11,-\frac{1}{7},\frac{5}{8}\right)$
 $\frac{{409124992664884260142733}}{{27119645385619965562847232}}$ (4)

Using the alternate definition for the Laguerre polynomials:

 > $\mathrm{altL}≔\left(n,a,x\right)→{\left(-1\right)}^{a}n!{\mathrm{orthopoly}}_{L}\left(n-a,a,x\right):$
 > $\mathrm{altL}\left(3,1,x\right)$
 ${-}{3}{}{{x}}^{{2}}{+}{18}{}{x}{-}{18}$ (5)

 See Also