Laguerre

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Laguerre

Hypergeometric representation

L n α ⁡ ( x ) = ( α + 1 ) n n ! ⁢ \HyperpFq ⁢ 11 ⁢ @ ⁢ @ - n ⁢ α + 1 ⁢ x generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 Pochhammer-symbol 𝛼 1 𝑛 𝑛 \HyperpFq 11 @ @ 𝑛 𝛼 1 𝑥 {\displaystyle{\displaystyle{\displaystyle L^{\alpha}_{n}\left(x\right)=\frac{% {\left(\alpha+1\right)_{n}}}{n!}\,\HyperpFq{1}{1}@@{-n}{\alpha+1}{x}}}} {\displaystyle \Laguerre[\alpha]{n}@{x}=\frac{\pochhammer{\alpha+1}{n}}{n!}\,\HyperpFq{1}{1}@@{-n}{\alpha+1}{x} }

Orthogonality relation(s)

∫ 0 ∞ e - x ⁢ x α ⁢ L m α ⁡ ( x ) ⁢ L n α ⁡ ( x ) ⁢ 𝑑 x = Γ ⁡ ( n + α + 1 ) n ! ⁢ δ m , n superscript subscript 0 𝑥 superscript 𝑥 𝛼 generalized-Laguerre-polynomial-L 𝛼 𝑚 𝑥 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 differential-d 𝑥 Euler-Gamma 𝑛 𝛼 1 𝑛 Kronecker-delta 𝑚 𝑛 {\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}}}} {\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): α > - 1 𝛼 1 {\displaystyle{\displaystyle{\displaystyle\alpha>-1}}}


Recurrence relation

( n + 1 ) ⁢ L n + 1 α ⁡ ( x ) - ( 2 ⁢ n + α + 1 - x ) ⁢ L n α ⁡ ( x ) + ( n + α ) ⁢ L n - 1 α ⁡ ( x ) = 0 𝑛 1 generalized-Laguerre-polynomial-L 𝛼 𝑛 1 𝑥 2 𝑛 𝛼 1 𝑥 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑛 𝛼 generalized-Laguerre-polynomial-L 𝛼 𝑛 1 𝑥 0 {\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}}} {\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

x ⁢ L ^ n ( α ) ⁡ ( x ) ⁢ x = L ^ n + 1 ( α ) ⁡ ( x ) ⁢ x + ( 2 ⁢ n + α + 1 ) ⁢ L ^ n ( α ) ⁡ ( x ) ⁢ x + n ⁢ ( n + α ) ⁢ L ^ n - 1 ( α ) ⁡ ( x ) ⁢ x 𝑥 generalized-Laguerre-polynomial-monic-p 𝛼 𝑛 𝑥 𝑥 generalized-Laguerre-polynomial-monic-p 𝛼 𝑛 1 𝑥 𝑥 2 𝑛 𝛼 1 generalized-Laguerre-polynomial-monic-p 𝛼 𝑛 𝑥 𝑥 𝑛 𝑛 𝛼 generalized-Laguerre-polynomial-monic-p 𝛼 𝑛 1 𝑥 𝑥 {\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 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} }
L n α ⁡ ( x ) = ( - 1 ) n n ! ⁢ L ^ n ( α ) ⁡ ( x ) ⁢ x generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 superscript 1 𝑛 𝑛 generalized-Laguerre-polynomial-monic-p 𝛼 𝑛 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle L^{\alpha}_{n}\left(x\right)=\frac{% (-1)^{n}}{n!}{\widehat{L}}^{(\alpha)}_{n}\left(x\right){x}}}} {\displaystyle \Laguerre[\alpha]{n}@{x}=\frac{(-1)^n}{n!}\monicLaguerre[\alpha]{n}@@{x}{x} }

Differential equation

x ⁢ y ′′ ⁢ ( x ) + ( α + 1 - x ) ⁢ y ′ ⁢ ( x ) + n ⁢ y ⁢ ( x ) = 0 𝑥 superscript 𝑦 ′′ 𝑥 𝛼 1 𝑥 superscript 𝑦 ′ 𝑥 𝑛 𝑦 𝑥 0 {\displaystyle{\displaystyle{\displaystyle xy^{\prime\prime}(x)+(\alpha+1-x)y^% {\prime}(x)+ny(x)=0}}} {\displaystyle xy''(x)+(\alpha+1-x)y'(x)+ny(x)=0 }

Substitution(s): y ⁢ ( x ) = L n α ⁡ ( x ) 𝑦 𝑥 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle y(x)=L^{\alpha}_{n}\left(x\right)}}}


Forward shift operator

d d ⁢ x ⁢ L n α ⁡ ( x ) = - L n - 1 α + 1 ⁡ ( x ) 𝑑 𝑑 𝑥 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 generalized-Laguerre-polynomial-L 𝛼 1 𝑛 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}L^{\alpha}_{n}\left(x% \right)=-L^{\alpha+1}_{n-1}\left(x\right)}}} {\displaystyle \frac{d}{dx}\Laguerre[\alpha]{n}@{x}=-\Laguerre[\alpha+1]{n-1}@{x} }

Backward shift operator

x ⁢ d d ⁢ x ⁢ L n α ⁡ ( x ) + ( α - x ) ⁢ L n α ⁡ ( x ) = ( n + 1 ) ⁢ L n + 1 α - 1 ⁡ ( x ) 𝑥 𝑑 𝑑 𝑥 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝛼 𝑥 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑛 1 generalized-Laguerre-polynomial-L 𝛼 1 𝑛 1 𝑥 {\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 x\frac{d}{dx}\Laguerre[\alpha]{n}@{x}+(\alpha-x)\Laguerre[\alpha]{n}@{x}=(n+1)\Laguerre[\alpha-1]{n+1}@{x} }
d d ⁢ x ⁢ [ e - x ⁢ x α ⁢ L n α ⁡ ( x ) ] = ( n + 1 ) ⁢ e - x ⁢ x α - 1 ⁢ L n + 1 ( α - 1 ) ⁢ ( x ) 𝑑 𝑑 𝑥 delimited-[] 𝑥 superscript 𝑥 𝛼 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑛 1 𝑥 superscript 𝑥 𝛼 1 subscript superscript 𝐿 𝛼 1 𝑛 1 𝑥 {\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)}}} {\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

e - x ⁢ x α ⁢ L n α ⁡ ( x ) = 1 n ! ⁢ ( d d ⁢ x ) n ⁢ [ e - x ⁢ x n + α ] 𝑥 superscript 𝑥 𝛼 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 1 𝑛 superscript 𝑑 𝑑 𝑥 𝑛 delimited-[] 𝑥 superscript 𝑥 𝑛 𝛼 {\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]}}} {\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

( 1 - t ) - α - 1 ⁢ exp ⁡ ( x ⁢ t t - 1 ) = ∑ n = 0 ∞ L n α ⁡ ( x ) ⁢ t n superscript 1 𝑡 𝛼 1 𝑥 𝑡 𝑡 1 superscript subscript 𝑛 0 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 superscript 𝑡 𝑛 {\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 (1-t)^{-\alpha-1}\exp@{\frac{xt}{t-1}}= \sum_{n=0}^{\infty}\Laguerre[\alpha]{n}@{x}t^n }
e t ⁢ \HyperpFq ⁢ 01 ⁢ @ ⁢ @ - α + 1 - x ⁢ t = ∑ n = 0 ∞ L n α ⁡ ( x ) ( α + 1 ) n ⁢ t n 𝑡 \HyperpFq 01 @ @ 𝛼 1 𝑥 𝑡 superscript subscript 𝑛 0 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 Pochhammer-symbol 𝛼 1 𝑛 superscript 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 \expe^t\,\HyperpFq{0}{1}@@{-}{\alpha+1}{-xt} =\sum_{n=0}^{\infty}\frac{\Laguerre[\alpha]{n}@{x}}{\pochhammer{\alpha+1}{n}}t^n }
( 1 - t ) - γ ⁢ \HyperpFq ⁢ 11 ⁢ @ ⁢ @ ⁢ γ ⁢ α + 1 ⁢ x ⁢ t t - 1 = ∑ n = 0 ∞ ( γ ) n ( α + 1 ) n ⁢ L n α ⁡ ( x ) ⁢ t n superscript 1 𝑡 𝛾 \HyperpFq 11 @ @ 𝛾 𝛼 1 𝑥 𝑡 𝑡 1 superscript subscript 𝑛 0 Pochhammer-symbol 𝛾 𝑛 Pochhammer-symbol 𝛼 1 𝑛 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 superscript 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}}}} {\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): γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


Limit relations

Meixner-Pollaczek polynomial to Laguerre polynomial

lim ϕ → 0 ⁡ P n ( 1 2 ⁢ α + 1 2 ) ⁡ ( - 1 2 ⁢ ϕ - 1 ⁢ x ; ϕ ) = L n α ⁡ ( x ) subscript → italic-ϕ 0 Meixner-Pollaczek-polynomial-P 1 2 𝛼 1 2 𝑛 1 2 superscript italic-ϕ 1 𝑥 italic-ϕ generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\lim_{\phi\rightarrow 0}P^{(\frac{1}% {2}\alpha+\frac{1}{2})}_{n}\!\left(-\textstyle\frac{1}{2}\phi^{-1}x;\phi\right% )=L^{\alpha}_{n}\left(x\right)}}} {\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

lim β → ∞ ⁡ P n ( α , β ) ⁡ ( 1 - 2 ⁢ β - 1 ⁢ x ) = L n α ⁡ ( x ) subscript → 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 2 superscript 𝛽 1 𝑥 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\lim_{\beta\rightarrow\infty}P^{(% \alpha,\beta)}_{n}\left(1-2\beta^{-1}x\right)=L^{\alpha}_{n}\left(x\right)}}} {\displaystyle \lim_{\beta\rightarrow\infty} \Jacobi{\alpha}{\beta}{n}@{1-2\beta^{-1}x}=\Laguerre[\alpha]{n}@{x} }

Meixner polynomial to Laguerre polynomial

lim c → 1 ⁡ M n ⁡ ( ( 1 - c ) - 1 ⁢ x ; α + 1 , c ) = L n α ⁡ ( x ) L n α ⁡ ( 0 ) subscript → 𝑐 1 Meixner-polynomial-M 𝑛 superscript 1 𝑐 1 𝑥 𝛼 1 𝑐 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 generalized-Laguerre-polynomial-L 𝛼 𝑛 0 {\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)}}}} {\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

lim α → ∞ ( 2 α ) 1 2 ⁢ n L n α ⁡ ( ( 2 α ) 1 2 x + α ) = ( - 1 ) n n ! H n ⁡ ( x ) fragments subscript → 𝛼 superscript fragments ( 2 𝛼 ) 1 2 𝑛 superscript generalized-Laguerre-polynomial-L 𝛼 𝑛 fragments ( 2 α 1 2 x α ) superscript 1 𝑛 𝑛 Hermite-polynomial-H 𝑛 𝑥 {\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)}}} {\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

L n α ⁡ ( x ) = 1 n ! ⁢ ∑ k = 0 n ( - n ) k k ! ⁢ ( α + k + 1 ) n - k ⁢ x k generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 1 𝑛 superscript subscript 𝑘 0 𝑛 Pochhammer-symbol 𝑛 𝑘 𝑘 Pochhammer-symbol 𝛼 𝑘 1 𝑛 𝑘 superscript 𝑥 𝑘 {\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 \Laguerre[\alpha]{n}@{x}=\frac{1}{n!}\sum_{k=0}^n\frac{\pochhammer{-n}{k}}{k!}\pochhammer{\alpha+k+1}{n-k}x^k }
L n α ⁡ ( x ) = ( - x ) n n ! ⁢ y n ⁡ ( 2 ⁢ x - 1 ; - 2 ⁢ n - α - 1 ) generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 superscript 𝑥 𝑛 𝑛 Bessel-polynomial-y 𝑛 2 superscript 𝑥 1 2 𝑛 𝛼 1 {\displaystyle{\displaystyle{\displaystyle L^{\alpha}_{n}\left(x\right)=\frac{% (-x)^{n}}{n!}y_{n}\!\left(2x^{-1};-2n-\alpha-1\right)}}} {\displaystyle \Laguerre[\alpha]{n}@{x}=\frac{(-x)^n}{n!}\BesselPoly{n}@{2x^{-1}}{-2n-\alpha-1} }
( - a ) n n ! ⁢ C n ⁡ ( x ; a ) = L n x - n ⁡ ( a ) superscript 𝑎 𝑛 𝑛 Charlier-polynomial-C 𝑛 𝑥 𝑎 generalized-Laguerre-polynomial-L 𝑥 𝑛 𝑛 𝑎 {\displaystyle{\displaystyle{\displaystyle\frac{(-a)^{n}}{n!}C_{n}\!\left(x;a% \right)=L^{x-n}_{n}\left(a\right)}}} {\displaystyle \frac{(-a)^n}{n!}\Charlier{n}@{x}{a}=\Laguerre[x-n]{n}@{a} }
H 2 ⁢ n ⁡ ( x ) = ( - 1 ) n ⁢ n ! ⁢  2 2 ⁢ n ⁢ L n - 1 2 ⁡ ( x 2 ) Hermite-polynomial-H 2 𝑛 𝑥 superscript 1 𝑛 𝑛 superscript  2 2 𝑛 generalized-Laguerre-polynomial-L 1 2 𝑛 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle H_{2n}\left(x\right)=(-1)^{n}n!\,2^% {2n}L^{-\frac{1}{2}}_{n}\left(x^{2}\right)}}} {\displaystyle \Hermite{2n}@{x}=(-1)^nn!\,2^{2n}\Laguerre[-\frac{1}{2}]{n}@{x^2} }
H 2 ⁢ n + 1 ⁡ ( x ) = ( - 1 ) n ⁢ n ! ⁢  2 2 ⁢ n + 1 ⁢ x ⁢ L n 1 2 ⁡ ( x 2 ) Hermite-polynomial-H 2 𝑛 1 𝑥 superscript 1 𝑛 𝑛 superscript  2 2 𝑛 1 𝑥 generalized-Laguerre-polynomial-L 1 2 𝑛 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle H_{2n+1}\left(x\right)=(-1)^{n}n!\,% 2^{2n+1}xL^{\frac{1}{2}}_{n}\left(x^{2}\right)}}} {\displaystyle \Hermite{2n+1}@{x}=(-1)^nn!\,2^{2n+1}x\Laguerre[\frac{1}{2}]{n}@{x^2} }

Koornwinder Addendum: Laguerre

Laguerre: Special value

L n α ⁡ ( 0 ) = ( α + 1 ) n n ! generalized-Laguerre-polynomial-L 𝛼 𝑛 0 Pochhammer-symbol 𝛼 1 𝑛 𝑛 {\displaystyle{\displaystyle{\displaystyle L^{\alpha}_{n}\left(0\right)=\frac{% {\left(\alpha+1\right)_{n}}}{n!}}}} {\displaystyle \Laguerre[\alpha]{n}@{0}=\frac{\pochhammer{\alpha+1}{n}}{n!} }

Quadratic transformations

H 2 ⁢ n ⁡ ( x ) = ( - 1 ) n ⁢ 2 2 ⁢ n ⁢ n ! ⁢ L n - 1 / 2 ⁡ ( x 2 ) Hermite-polynomial-H 2 𝑛 𝑥 superscript 1 𝑛 superscript 2 2 𝑛 𝑛 generalized-Laguerre-polynomial-L 1 2 𝑛 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle H_{2n}\left(x\right)=(-1)^{n}2^{2n}% n!L^{-1/2}_{n}\left(x^{2}\right)}}} {\displaystyle \Hermite{2n}@{x}=(-1)^n 2^{2n} n! \Laguerre[-1/2]{n}@{x^2} }
H 2 ⁢ n + 1 ⁡ ( x ) = ( - 1 ) n ⁢ 2 2 ⁢ n + 1 ⁢ n ! ⁢ x ⁢ L n 1 / 2 ⁡ ( x 2 ) Hermite-polynomial-H 2 𝑛 1 𝑥 superscript 1 𝑛 superscript 2 2 𝑛 1 𝑛 𝑥 generalized-Laguerre-polynomial-L 1 2 𝑛 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle H_{2n+1}\left(x\right)=(-1)^{n}2^{2% n+1}n!xL^{1/2}_{n}\left(x^{2}\right)}}} {\displaystyle \Hermite{2n+1}@{x}=(-1)^n 2^{2n+1} n! x \Laguerre[1/2]{n}@{x^2} }

Fourier transform

1 Γ ⁡ ( α + 1 ) ⁢ ∫ 0 ∞ L n α ⁡ ( y ) L n α ⁡ ( 0 ) ⁢ e - y ⁢ y α ⁢ e i ⁢ x ⁢ y ⁢ 𝑑 y = i n ⁢ y n ( i ⁢ y + 1 ) n + α + 1 1 Euler-Gamma 𝛼 1 superscript subscript 0 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑦 generalized-Laguerre-polynomial-L 𝛼 𝑛 0 𝑦 superscript 𝑦 𝛼 imaginary-unit 𝑥 𝑦 differential-d 𝑦 imaginary-unit 𝑛 superscript 𝑦 𝑛 superscript imaginary-unit 𝑦 1 𝑛 𝛼 1 {\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}}}}} {\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

d d ⁢ x ⁢ ( x α ⁢ L n α ⁡ ( x ) ) = ( n + α ) ⁢ x α - 1 ⁢ L n α - 1 ⁡ ( x ) , ( x ⁢ d d ⁢ x + α ) ⁢ L n α ⁡ ( x ) 𝑑 𝑑 𝑥 superscript 𝑥 𝛼 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑛 𝛼 superscript 𝑥 𝛼 1 generalized-Laguerre-polynomial-L 𝛼 1 𝑛 𝑥 𝑥 𝑑 𝑑 𝑥 𝛼 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 {\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 \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} }
d d ⁢ x ⁢ ( x α ⁢ L n α ⁡ ( x ) ) = ( n + α ) ⁢ L n α - 1 ⁡ ( x ) 𝑑 𝑑 𝑥 superscript 𝑥 𝛼 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑛 𝛼 generalized-Laguerre-polynomial-L 𝛼 1 𝑛 𝑥 {\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 \frac d{dx}\left(x^\alpha \Laguerre[\alpha]{n}@{x}\right) = (n+\alpha) \Laguerre[\alpha-1]{n}@{x} }
d d ⁢ x ⁢ ( e - x ⁢ L n α ⁡ ( x ) ) = - e - x ⁢ L n α + 1 ⁡ ( x ) , ( d d ⁢ x - 1 ) ⁢ L n α ⁡ ( x ) 𝑑 𝑑 𝑥 𝑥 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑥 generalized-Laguerre-polynomial-L 𝛼 1 𝑛 𝑥 𝑑 𝑑 𝑥 1 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 {\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 \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} }
d d ⁢ x ⁢ ( e - x ⁢ L n α ⁡ ( x ) ) = - L n α + 1 ⁡ ( x ) 𝑑 𝑑 𝑥 𝑥 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 generalized-Laguerre-polynomial-L 𝛼 1 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left({\mathrm{e}^{-x}}L% ^{\alpha}_{n}\left(x\right)\right)=-L^{\alpha+1}_{n}\left(x\right)}}} {\displaystyle \frac d{dx}\left(\expe^{-x} \Laguerre[\alpha]{n}@{x}\right) = -\Laguerre[\alpha+1]{n}@{x} }

Generalized Hermite polynomials

H 2 ⁢ m μ ⁡ ( x ) := c ⁢ o ⁢ n ⁢ s ⁢ t ⁢ L m μ - 1 2 ⁡ ( x 2 ) , H 2 ⁢ m + 1 μ ⁡ ( x ) : : assign generalized-Hermite-polynomial-H 𝜇 2 𝑚 𝑥 c 𝑜 𝑛 𝑠 𝑡 generalized-Laguerre-polynomial-L 𝜇 1 2 𝑚 superscript 𝑥 2 generalized-Hermite-polynomial-H 𝜇 2 𝑚 1 𝑥 absent {\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 \GenHermite[\mu]{2m}@{x}:=\mathrm const \Laguerre[\mu-\frac12]{m}@{x^2},\qquad \GenHermite[\mu]{2m+1}@{x}: }
H 2 ⁢ m μ ⁡ ( x ) = c ⁢ o ⁢ n ⁢ s ⁢ t ⁢ x ⁢ L m μ + 1 2 ⁡ ( x 2 ) generalized-Hermite-polynomial-H 𝜇 2 𝑚 𝑥 c 𝑜 𝑛 𝑠 𝑡 𝑥 generalized-Laguerre-polynomial-L 𝜇 1 2 𝑚 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle H^{\mu}_{2m}\left(x\right)=\mathrm{% c}onstxL^{\mu+\frac{1}{2}}_{m}\left(x^{2}\right)}}} {\displaystyle \GenHermite[\mu]{2m}@{x} =\mathrm const x \Laguerre[\mu+\frac12]{m}@{x^2} }
∫ - ∞ ∞ H m μ ⁡ ( x ) H n μ ⁡ ( x ) | x | 2 ⁢ μ e - x 2 d x = 0    ( m ≠ n ) fragments superscript subscript generalized-Hermite-polynomial-H 𝜇 𝑚 𝑥 generalized-Hermite-polynomial-H 𝜇 𝑛 𝑥 | x superscript | 2 𝜇 superscript 𝑥 2 d x 0 italic-   fragments ( m n ) {\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 \int_{-\infty}^{\infty} \GenHermite[\mu]{m}@{x} \GenHermite[\mu]{n}@{x} |x|^{2\mu}\expe^{-x^2} dx =0\qquad(m\ne n) }
H 2 ⁢ m μ ⁡ ( x ) = ( - 1 ) m ⁢ ( 2 ⁢ m ) ! ( μ + 1 2 ) m ⁢ L m μ - 1 2 ⁡ ( x 2 ) , H 2 ⁢ m + 1 μ ⁡ ( x ) generalized-Hermite-polynomial-H 𝜇 2 𝑚 𝑥 superscript 1 𝑚 2 𝑚 Pochhammer-symbol 𝜇 1 2 𝑚 generalized-Laguerre-polynomial-L 𝜇 1 2 𝑚 superscript 𝑥 2 generalized-Hermite-polynomial-H 𝜇 2 𝑚 1 𝑥 {\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 \GenHermite[\mu]{2m}@{x}=\frac{(-1)^m(2m)!}{\pochhammer{\mu+\frac12}{m}} \Laguerre[\mu-\frac12]{m}@{x^2},\qquad \GenHermite[\mu]{2m+1}@{x} }
H 2 ⁢ m μ ⁡ ( x ) = ( - 1 ) m ⁢ ( 2 ⁢ m + 1 ) ! ( μ + 1 2 ) m + 1 ⁢ x ⁢ L m μ + 1 2 ⁡ ( x 2 ) generalized-Hermite-polynomial-H 𝜇 2 𝑚 𝑥 superscript 1 𝑚 2 𝑚 1 Pochhammer-symbol 𝜇 1 2 𝑚 1 𝑥 generalized-Laguerre-polynomial-L 𝜇 1 2 𝑚 superscript 𝑥 2 {\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 \GenHermite[\mu]{2m}@{x} =\frac{(-1)^m(2m+1)!}{\pochhammer{\mu+\frac12}{m+1}} x \Laguerre[\mu+\frac12]{m}@{x^2} \label{80} }
T μ ⁢ H n μ = 2 ⁢ n ⁢ H n - 1 μ subscript 𝑇 𝜇 superscript subscript 𝐻 𝑛 𝜇 2 𝑛 superscript subscript 𝐻 𝑛 1 𝜇 {\displaystyle{\displaystyle{\displaystyle T_{\mu}H_{n}^{\mu}=2nH_{n-1}^{\mu}}}} {\displaystyle T_\mu H_n^\mu=2n H_{n-1}^\mu }
T μ 2 ⁢ H n μ = 4 ⁢ n ⁢ ( n - 1 ) ⁢ H n - 2 μ superscript subscript 𝑇 𝜇 2 superscript subscript 𝐻 𝑛 𝜇 4 𝑛 𝑛 1 superscript subscript 𝐻 𝑛 2 𝜇 {\displaystyle{\displaystyle{\displaystyle T_{\mu}^{2}H_{n}^{\mu}=4n(n-1)H_{n-% 2}^{\mu}}}} {\displaystyle T_\mu^2 H_n^\mu=4n(n-1) H_{n-2}^\mu }
( d 2 d ⁢ x 2 + 2 ⁢ α + 1 x ⁢ d d ⁢ x ) ⁢ L n α ⁡ ( x 2 ) = - 4 ⁢ ( n + α ) ⁢ L n - 1 α ⁡ ( x 2 ) superscript 𝑑 2 𝑑 superscript 𝑥 2 2 𝛼 1 𝑥 𝑑 𝑑 𝑥 generalized-Laguerre-polynomial-L 𝛼 𝑛 superscript 𝑥 2 4 𝑛 𝛼 generalized-Laguerre-polynomial-L 𝛼 𝑛 1 superscript 𝑥 2 {\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)}}} {\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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