Laguerre

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Laguerre

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Hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle L^{\alpha}_{n}\left(x\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}\,\HyperpFq{1}{1}@@{-n}{\alpha+1}{x}}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\int_{0}^{\infty}{\mathrm{e}^{-x}}x^{\alpha}L^{\alpha}_{m}\left(x\right)L^{\alpha}_{n}\left(x\right)\,dx=\frac{\Gamma\left(n+\alpha+1\right)}{n!}\,\delta_{m,n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\alpha>-1}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle(n+1)L^{\alpha}_{n+1}\left(x\right)-(2n+\alpha+1-x)L^{\alpha}_{n}\left(x\right)+(n+\alpha)L^{\alpha}_{n-1}\left(x\right)=0}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{L}}^{(\alpha)}_{n}\left(x\right){x}={\widehat{L}}^{(\alpha)}_{n+1}\left(x\right){x}+(2n+\alpha+1){\widehat{L}}^{(\alpha)}_{n}\left(x\right){x}+n(n+\alpha){\widehat{L}}^{(\alpha)}_{n-1}\left(x\right){x}}}}$
${\displaystyle{\displaystyle{\displaystyle L^{\alpha}_{n}\left(x\right)=\frac{(-1)^{n}}{n!}{\widehat{L}}^{(\alpha)}_{n}\left(x\right){x}}}}$

Differential equation

${\displaystyle{\displaystyle{\displaystyle xy^{\prime\prime}(x)+(\alpha+1-x)y^{\prime}(x)+ny(x)=0}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle y(x)=L^{\alpha}_{n}\left(x\right)}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}L^{\alpha}_{n}\left(x\right)=-L^{\alpha+1}_{n-1}\left(x\right)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle x\frac{d}{dx}L^{\alpha}_{n}\left(x\right)+(\alpha-x)L^{\alpha}_{n}\left(x\right)=(n+1)L^{\alpha-1}_{n+1}\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left[{\mathrm{e}^{-x}}x^{\alpha}L^{\alpha}_{n}\left(x\right)\right]=(n+1){\mathrm{e}^{-x}}x^{\alpha-1}L^{(\alpha-1)}_{n+1}(x)}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{-x}}x^{\alpha}L^{\alpha}_{n}\left(x\right)=\frac{1}{n!}\left(\frac{d}{dx}\right)^{n}\left[{\mathrm{e}^{-x}}x^{n+\alpha}\right]}}}$

Generating functions

${\displaystyle{\displaystyle{\displaystyle(1-t)^{-\alpha-1}\exp\left(\frac{xt}{t-1}\right)=\sum_{n=0}^{\infty}L^{\alpha}_{n}\left(x\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{t}}\,\HyperpFq{0}{1}@@{-}{\alpha+1}{-xt}=\sum_{n=0}^{\infty}\frac{L^{\alpha}_{n}\left(x\right)}{{\left(\alpha+1\right)_{n}}}t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle(1-t)^{-\gamma}\,\HyperpFq{1}{1}@@{\gamma}{\alpha+1}{\frac{xt}{t-1}}=\sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}}}{{\left(\alpha+1\right)_{n}}}L^{\alpha}_{n}\left(x\right)t^{n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\gamma}}}$ arbitrary

Limit relations

Meixner-Pollaczek polynomial to Laguerre polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\phi\rightarrow 0}\MeixnerPollaczek{\frac{1}{2}\alpha+\frac{1}{2}}{n}@{-\textstyle\frac{1}{2}\phi^{-1}x}{\phi}=\Laguerre[\alpha]{n}@{x}}}}$

Jacobi polynomial to Laguerre polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\beta\rightarrow\infty}P^{(\alpha,\beta)}_{n}\left(1-2\beta^{-1}x\right)=L^{\alpha}_{n}\left(x\right)}}}$

Meixner polynomial to Laguerre polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{c\rightarrow 1}M_{n}\!\left((1-c)^{-1}x;\alpha+1,c\right)=\frac{L^{\alpha}_{n}\left(x\right)}{L^{\alpha}_{n}\left(0\right)}}}}$

Laguerre polynomial to Hermite polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n}L^{\alpha}_{n}\left((2\alpha\right)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^{n}}{n!}H_{n}\left(x\right)}}}$

Remarks

${\displaystyle{\displaystyle{\displaystyle L^{\alpha}_{n}\left(x\right)=\frac{1}{n!}\sum_{k=0}^{n}\frac{{\left(-n\right)_{k}}}{k!}{\left(\alpha+k+1\right)_{n-k}}x^{k}}}}$
${\displaystyle{\displaystyle{\displaystyle L^{\alpha}_{n}\left(x\right)=\frac{(-x)^{n}}{n!}y_{n}\!\left(2x^{-1};-2n-\alpha-1\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{(-a)^{n}}{n!}C_{n}\!\left(x;a\right)=L^{x-n}_{n}\left(a\right)}}}$
${\displaystyle{\displaystyle{\displaystyle H_{2n}\left(x\right)=(-1)^{n}n!\,2^{2n}L^{-\frac{1}{2}}_{n}\left(x^{2}\right)}}}$
${\displaystyle{\displaystyle{\displaystyle H_{2n+1}\left(x\right)=(-1)^{n}n!\,2^{2n+1}xL^{\frac{1}{2}}_{n}\left(x^{2}\right)}}}$

Koornwinder Addendum: Laguerre

Laguerre: Special value

${\displaystyle{\displaystyle{\displaystyle L^{\alpha}_{n}\left(0\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}}}}$

Quadratic transformations

${\displaystyle{\displaystyle{\displaystyle H_{2n}\left(x\right)=(-1)^{n}2^{2n}n!L^{-1/2}_{n}\left(x^{2}\right)}}}$
${\displaystyle{\displaystyle{\displaystyle H_{2n+1}\left(x\right)=(-1)^{n}2^{2n+1}n!xL^{1/2}_{n}\left(x^{2}\right)}}}$

Fourier transform

${\displaystyle{\displaystyle{\displaystyle\frac{1}{\Gamma\left(\alpha+1\right)}\int_{0}^{\infty}\frac{L^{\alpha}_{n}\left(y\right)}{L^{\alpha}_{n}\left(0\right)}{\mathrm{e}^{-y}}y^{\alpha}{\mathrm{e}^{\mathrm{i}xy}}dy={\mathrm{i}^{n}}\frac{y^{n}}{(\mathrm{i}y+1)^{n+\alpha+1}}}}}$

Differentiation formulas

${\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left(x^{\alpha}L^{\alpha}_{n}\left(x\right)\right)=(n+\alpha)x^{\alpha-1}L^{\alpha-1}_{n}\left(x\right),\qquad\left(x\frac{d}{dx}+\alpha\right)L^{\alpha}_{n}\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left(x^{\alpha}L^{\alpha}_{n}\left(x\right)\right)=(n+\alpha)L^{\alpha-1}_{n}\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left({\mathrm{e}^{-x}}L^{\alpha}_{n}\left(x\right)\right)=-{\mathrm{e}^{-x}}L^{\alpha+1}_{n}\left(x\right),\qquad\left(\frac{d}{dx}-1\right)L^{\alpha}_{n}\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left({\mathrm{e}^{-x}}L^{\alpha}_{n}\left(x\right)\right)=-L^{\alpha+1}_{n}\left(x\right)}}}$

Generalized Hermite polynomials

${\displaystyle{\displaystyle{\displaystyle H^{\mu}_{2m}\left(x\right):=\mathrm{c}onstL^{\mu-\frac{1}{2}}_{m}\left(x^{2}\right),\qquad H^{\mu}_{2m+1}\left(x\right):}}}$
${\displaystyle{\displaystyle{\displaystyle H^{\mu}_{2m}\left(x\right)=\mathrm{c}onstxL^{\mu+\frac{1}{2}}_{m}\left(x^{2}\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\int_{-\infty}^{\infty}H^{\mu}_{m}\left(x\right)H^{\mu}_{n}\left(x\right)|x|^{2\mu}{\mathrm{e}^{-x^{2}}}dx=0\qquad(m\neq n)}}}$
${\displaystyle{\displaystyle{\displaystyle H^{\mu}_{2m}\left(x\right)=\frac{(-1)^{m}(2m)!}{{\left(\mu+\frac{1}{2}\right)_{m}}}L^{\mu-\frac{1}{2}}_{m}\left(x^{2}\right),\qquad H^{\mu}_{2m+1}\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle H^{\mu}_{2m}\left(x\right)=\frac{(-1)^{m}(2m+1)!}{{\left(\mu+\frac{1}{2}\right)_{m+1}}}xL^{\mu+\frac{1}{2}}_{m}\left(x^{2}\right)}}}$
${\displaystyle{\displaystyle{\displaystyle T_{\mu}H_{n}^{\mu}=2nH_{n-1}^{\mu}}}}$
${\displaystyle{\displaystyle{\displaystyle T_{\mu}^{2}H_{n}^{\mu}=4n(n-1)H_{n-2}^{\mu}}}}$
${\displaystyle{\displaystyle{\displaystyle\left(\frac{d^{2}}{dx^{2}}+\frac{2\alpha+1}{x}\frac{d}{dx}\right)L^{\alpha}_{n}\left(x^{2}\right)=-4(n+\alpha)L^{\alpha}_{n-1}\left(x^{2}\right)}}}$

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Laguerre

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