Laguerre
Hypergeometric representation
Orthogonality relation(s)
Recurrence relation
Monic recurrence relation
![{\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}
}}](/index.php?title=Special:MathShowImage&hash=9d1a5ab1e281591a0aeb2b4a335850fd&mode=latexml)
Differential equation
Forward shift operator
Backward shift operator
![{\displaystyle {\displaystyle
x\frac{d}{dx}\Laguerre[\alpha]{n}@{x}+(\alpha-x)\Laguerre[\alpha]{n}@{x}=(n+1)\Laguerre[\alpha-1]{n+1}@{x}
}}](/index.php?title=Special:MathShowImage&hash=d604f9057f698154032f7203abbebe36&mode=latexml)
Rodrigues-type formula
Generating functions
![{\displaystyle {\displaystyle
(1-t)^{-\alpha-1}\exp@{\frac{xt}{t-1}}=
\sum_{n=0}^{\infty}\Laguerre[\alpha]{n}@{x}t^n
}}](/index.php?title=Special:MathShowImage&hash=9ce6c737c3d5ca9bf4eae4e13db57bad&mode=latexml)
![{\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
}}](/index.php?title=Special:MathShowImage&hash=951215c6a54ce77c796dfee9fdeb515e&mode=latexml)
Limit relations
Meixner-Pollaczek polynomial to Laguerre polynomial
![{\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}
}}](/index.php?title=Special:MathShowImage&hash=eba077e8c6dd14c73b813508b1118b76&mode=latexml)
Jacobi polynomial to Laguerre polynomial
![{\displaystyle {\displaystyle
\lim_{\beta\rightarrow\infty}
\Jacobi{\alpha}{\beta}{n}@{1-2\beta^{-1}x}=\Laguerre[\alpha]{n}@{x}
}}](/index.php?title=Special:MathShowImage&hash=451228549675e87bf330bcb6bf23655e&mode=latexml)
Meixner polynomial to Laguerre polynomial
![{\displaystyle {\displaystyle
\lim_{c\rightarrow 1}
\Meixner{n}@{(1-c)^{-1}x}{\alpha+1}{c}=\frac{\Laguerre[\alpha]{n}@{x}}{\Laguerre[\alpha]{n}@{0}}
}}](/index.php?title=Special:MathShowImage&hash=4bf8fe817b9486cc5ad408aaca2354e4&mode=latexml)
Laguerre polynomial to Hermite polynomial
![{\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
}}](/index.php?title=Special:MathShowImage&hash=b5af06db5901510dd5e03e0edc1573b4&mode=latexml)
![{\displaystyle {\displaystyle
\Laguerre[\alpha]{n}@{x}=\frac{(-x)^n}{n!}\BesselPoly{n}@{2x^{-1}}{-2n-\alpha-1}
}}](/index.php?title=Special:MathShowImage&hash=44c4ea52f48533c8d3e2fc99b6c8d4d8&mode=latexml)
![{\displaystyle {\displaystyle
\frac{(-a)^n}{n!}\Charlier{n}@{x}{a}=\Laguerre[x-n]{n}@{a}
}}](/index.php?title=Special:MathShowImage&hash=47732debda70b059ed4fcee9b182afeb&mode=latexml)
![{\displaystyle {\displaystyle
\Hermite{2n}@{x}=(-1)^nn!\,2^{2n}\Laguerre[-\frac{1}{2}]{n}@{x^2}
}}](/index.php?title=Special:MathShowImage&hash=779facce98fdd8dc13ce4fcb10c70ae4&mode=latexml)
Koornwinder Addendum: Laguerre
Laguerre: Special value
![{\displaystyle {\displaystyle
\Laguerre[\alpha]{n}@{0}=\frac{\pochhammer{\alpha+1}{n}}{n!}
}}](/index.php?title=Special:MathShowImage&hash=878948b4485ec6ee59ce9924b1d93274&mode=latexml)
Quadratic transformations
![{\displaystyle {\displaystyle
\Hermite{2n}@{x}=(-1)^n 2^{2n} n! \Laguerre[-1/2]{n}@{x^2}
}}](/index.php?title=Special:MathShowImage&hash=558ad4beefbba3655af3bd6480a74872&mode=latexml)
![{\displaystyle {\displaystyle
\Hermite{2n+1}@{x}=(-1)^n 2^{2n+1} n! x \Laguerre[1/2]{n}@{x^2}
}}](/index.php?title=Special:MathShowImage&hash=eba86698adbec5691e1e3a8be464cb75&mode=latexml)
Fourier transform
![{\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}}
}}](/index.php?title=Special:MathShowImage&hash=e5f2ff0b709ca1ab343d102bb9cb702c&mode=latexml)
Differentiation formulas
![{\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}
}}](/index.php?title=Special:MathShowImage&hash=490120d9940da0d1d3a0775ecab2e68f&mode=latexml)
![{\displaystyle {\displaystyle
\frac d{dx}\left(x^\alpha \Laguerre[\alpha]{n}@{x}\right)
=
(n+\alpha) \Laguerre[\alpha-1]{n}@{x}
}}](/index.php?title=Special:MathShowImage&hash=6ade108338c20b2fbd50d9ae85c18d45&mode=latexml)
![{\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}
}}](/index.php?title=Special:MathShowImage&hash=2914c7d7bf2ef0b83a052ef548520c81&mode=latexml)
![{\displaystyle {\displaystyle
\frac d{dx}\left(\expe^{-x} \Laguerre[\alpha]{n}@{x}\right)
=
-\Laguerre[\alpha+1]{n}@{x}
}}](/index.php?title=Special:MathShowImage&hash=fb3b406059c4cb1ce9d26bd6b8a6c4c0&mode=latexml)
Generalized Hermite polynomials
![{\displaystyle {\displaystyle
\GenHermite[\mu]{2m}@{x}:=\mathrm const \Laguerre[\mu-\frac12]{m}@{x^2},\qquad
\GenHermite[\mu]{2m+1}@{x}:
}}](/index.php?title=Special:MathShowImage&hash=aed40ca7330c5d270ad04cb38d5e55ac&mode=latexml)
![{\displaystyle {\displaystyle
\GenHermite[\mu]{2m}@{x}
=\mathrm const x \Laguerre[\mu+\frac12]{m}@{x^2}
}}](/index.php?title=Special:MathShowImage&hash=c3e147ecf2b969be43db732ea984d3db&mode=latexml)
![{\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)
}}](/index.php?title=Special:MathShowImage&hash=af2717fb19ab6d26eeba1cac9736d31b&mode=latexml)
![{\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}
}}](/index.php?title=Special:MathShowImage&hash=568aed698c2a09cd199687da7302dc33&mode=latexml)
![{\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}
}}](/index.php?title=Special:MathShowImage&hash=b2c6a6637cc0e41415e3ce390d29395c&mode=latexml)

