Laguerre

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Laguerre

Hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Laguerre[\alpha]{n}@{x}=\frac{\pochhammer{\alpha+1}{n}}{n!}\,\HyperpFq{1}{1}@@{-n}{\alpha+1}{x} }}

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_0^{\infty}\expe^{-x}x^{\alpha}\Laguerre[\alpha]{m}@{x}\Laguerre[\alpha]{n}@{x}\,dx= \frac{\EulerGamma@{n+\alpha+1}}{n!}\,\Kronecker{m}{n} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \alpha > -1}}


Recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (n+1)\Laguerre[\alpha]{n+1}@{x}-(2n+\alpha+1-x)\Laguerre[\alpha]{n}@{x}+(n+\alpha)\Laguerre[\alpha]{n-1}@{x}=0 }}

Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicLaguerre[\alpha]{n}@@{x}{x}=\monicLaguerre[\alpha]{n+1}@@{x}{x}+(2n+\alpha+1)\monicLaguerre[\alpha]{n}@@{x}{x}+n(n+\alpha)\monicLaguerre[\alpha]{n-1}@@{x}{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Laguerre[\alpha]{n}@{x}=\frac{(-1)^n}{n!}\monicLaguerre[\alpha]{n}@@{x}{x} }}

Differential equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle xy''(x)+(\alpha+1-x)y'(x)+ny(x)=0 }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\Laguerre[\alpha]{n}@{x}}}


Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{d}{dx}\Laguerre[\alpha]{n}@{x}=-\Laguerre[\alpha+1]{n-1}@{x} }}

Backward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\frac{d}{dx}\Laguerre[\alpha]{n}@{x}+(\alpha-x)\Laguerre[\alpha]{n}@{x}=(n+1)\Laguerre[\alpha-1]{n+1}@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{d}{dx}\left[\expe^{-x}x^{\alpha}\Laguerre[\alpha]{n}@{x}\right]=(n+1)\expe^{-x}x^{\alpha-1}L^{(\alpha-1)}_{n+1}(x) }}

Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \expe^{-x}x^{\alpha}\Laguerre[\alpha]{n}@{x}=\frac{1}{n!}\left(\frac{d}{dx}\right)^n\left[\expe^{-x}x^{n+\alpha}\right] }}

Generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-t)^{-\alpha-1}\exp@{\frac{xt}{t-1}}= \sum_{n=0}^{\infty}\Laguerre[\alpha]{n}@{x}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \expe^t\,\HyperpFq{0}{1}@@{-}{\alpha+1}{-xt} =\sum_{n=0}^{\infty}\frac{\Laguerre[\alpha]{n}@{x}}{\pochhammer{\alpha+1}{n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-t)^{-\gamma}\,\HyperpFq{1}{1}@@{\gamma}{\alpha+1}{\frac{xt}{t-1}} =\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}}{\pochhammer{\alpha+1}{n}}\Laguerre[\alpha]{n}@{x}t^n }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \gamma}} arbitrary


Limit relations

Meixner-Pollaczek polynomial to Laguerre polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\beta\rightarrow\infty} \Jacobi{\alpha}{\beta}{n}@{1-2\beta^{-1}x}=\Laguerre[\alpha]{n}@{x} }}

Meixner polynomial to Laguerre polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{c\rightarrow 1} \Meixner{n}@{(1-c)^{-1}x}{\alpha+1}{c}=\frac{\Laguerre[\alpha]{n}@{x}}{\Laguerre[\alpha]{n}@{0}} }}

Laguerre polynomial to Hermite polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\alpha\rightarrow\infty} \left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} \Laguerre[\alpha]{n}@{(2\alpha}^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}\Hermite{n}@{x} }}

Remarks

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Laguerre[\alpha]{n}@{x}=\frac{1}{n!}\sum_{k=0}^n\frac{\pochhammer{-n}{k}}{k!}\pochhammer{\alpha+k+1}{n-k}x^k }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Laguerre[\alpha]{n}@{x}=\frac{(-x)^n}{n!}\BesselPoly{n}@{2x^{-1}}{-2n-\alpha-1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{(-a)^n}{n!}\Charlier{n}@{x}{a}=\Laguerre[x-n]{n}@{a} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hermite{2n}@{x}=(-1)^nn!\,2^{2n}\Laguerre[-\frac{1}{2}]{n}@{x^2} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hermite{2n+1}@{x}=(-1)^nn!\,2^{2n+1}x\Laguerre[\frac{1}{2}]{n}@{x^2} }}

Koornwinder Addendum: Laguerre

Laguerre: Special value

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Laguerre[\alpha]{n}@{0}=\frac{\pochhammer{\alpha+1}{n}}{n!} }}

Quadratic transformations

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hermite{2n}@{x}=(-1)^n 2^{2n} n! \Laguerre[-1/2]{n}@{x^2} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hermite{2n+1}@{x}=(-1)^n 2^{2n+1} n! x \Laguerre[1/2]{n}@{x^2} }}

Fourier transform

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac1{\EulerGamma@{\alpha+1}} \int_0^\infty \frac{\Laguerre[\alpha]{n}@{y}}{\Laguerre[\alpha]{n}@{0}} \expe^{-y} y^\alpha \expe^{\iunit xy} dy= \iunit^n \frac{y^n}{(\iunit y+1)^{n+\alpha+1}} }}

Differentiation formulas

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac d{dx}\left(x^\alpha \Laguerre[\alpha]{n}@{x}\right)= (n+\alpha) x^{\alpha-1} \Laguerre[\alpha-1]{n}@{x},\qquad \left(x\frac d{dx}+\alpha\right)\Laguerre[\alpha]{n}@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac d{dx}\left(x^\alpha \Laguerre[\alpha]{n}@{x}\right) = (n+\alpha) \Laguerre[\alpha-1]{n}@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac d{dx}\left(\expe^{-x} \Laguerre[\alpha]{n}@{x}\right)= -\expe^{-x} \Laguerre[\alpha+1]{n}@{x},\qquad \left(\frac d{dx}-1\right)\Laguerre[\alpha]{n}@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac d{dx}\left(\expe^{-x} \Laguerre[\alpha]{n}@{x}\right) = -\Laguerre[\alpha+1]{n}@{x} }}

Generalized Hermite polynomials

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \GenHermite[\mu]{2m}@{x}:=\mathrm const \Laguerre[\mu-\frac12]{m}@{x^2},\qquad \GenHermite[\mu]{2m+1}@{x}: }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \GenHermite[\mu]{2m}@{x} =\mathrm const x \Laguerre[\mu+\frac12]{m}@{x^2} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_{-\infty}^{\infty} \GenHermite[\mu]{m}@{x} \GenHermite[\mu]{n}@{x} |x|^{2\mu}\expe^{-x^2} dx =0\qquad(m\ne n) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \GenHermite[\mu]{2m}@{x}=\frac{(-1)^m(2m)!}{\pochhammer{\mu+\frac12}{m}} \Laguerre[\mu-\frac12]{m}@{x^2},\qquad \GenHermite[\mu]{2m+1}@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \GenHermite[\mu]{2m}@{x} =\frac{(-1)^m(2m+1)!}{\pochhammer{\mu+\frac12}{m+1}} x \Laguerre[\mu+\frac12]{m}@{x^2} \label{80} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle T_\mu H_n^\mu=2n H_{n-1}^\mu }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle T_\mu^2 H_n^\mu=4n(n-1) H_{n-2}^\mu }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \left(\frac{d^2}{dx^2}+\frac{2\alpha+1}x \frac d{dx}\right)\Laguerre[\alpha]{n}@{x^2} =-4(n+\alpha) \Laguerre[\alpha]{n-1}@{x^2} }}

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