Hermite

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Hermite

Hypergeometric representation

H n ⁑ ( x ) = ( 2 x ) n \HyperpFq 20 @ @ - n / 2 , - ( n - 1 ) / 2 - - 1 x 2 fragments Hermite-polynomial-H 𝑛 π‘₯ superscript fragments ( 2 x ) 𝑛 \HyperpFq 20 @ @ n 2 , fragments ( n 1 ) 2 1 superscript π‘₯ 2 {\displaystyle{\displaystyle{\displaystyle H_{n}\left(x\right)=(2x)^{n}\,% \HyperpFq{2}{0}@@{-n/2,-(n-1)/2}{-}{-\frac{1}{x^{2}}}}}} {\displaystyle \Hermite{n}@{x}=(2x)^n\,\HyperpFq{2}{0}@@{-n/2,-(n-1)/2}{-}{-\frac{1}{x^2}} }

Orthogonality relation(s)

1 Ο€ ⁒ ∫ - ∞ ∞ e - x 2 ⁒ H m ⁑ ( x ) ⁒ H n ⁑ ( x ) ⁒ 𝑑 x = 2 n ⁒ n ! ⁒ Ξ΄ m , n 1 superscript subscript superscript π‘₯ 2 Hermite-polynomial-H π‘š π‘₯ Hermite-polynomial-H 𝑛 π‘₯ differential-d π‘₯ superscript 2 𝑛 𝑛 Kronecker-delta π‘š 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{\sqrt{\pi}}\int_{-\infty}^{% \infty}{\mathrm{e}^{-x^{2}}}H_{m}\left(x\right)H_{n}\left(x\right)\,dx=2^{n}n!% \,\delta_{m,n}}}} {\displaystyle \frac{1}{\sqrt{\cpi}}\int_{-\infty}^{\infty}\expe^{-x^2}\Hermite{m}@{x}\Hermite{n}@{x}\,dx =2^nn!\,\Kronecker{m}{n} }

Recurrence relation

H n + 1 ⁑ ( x ) - 2 ⁒ x ⁒ H n ⁑ ( x ) + 2 ⁒ n ⁒ H n - 1 ⁑ ( x ) = 0 Hermite-polynomial-H 𝑛 1 π‘₯ 2 π‘₯ Hermite-polynomial-H 𝑛 π‘₯ 2 𝑛 Hermite-polynomial-H 𝑛 1 π‘₯ 0 {\displaystyle{\displaystyle{\displaystyle H_{n+1}\left(x\right)-2xH_{n}\left(% x\right)+2nH_{n-1}\left(x\right)=0}}} {\displaystyle \Hermite{n+1}@{x}-2x\Hermite{n}@{x}+2n\Hermite{n-1}@{x}=0 }

Monic recurrence relation

x ⁒ H ^ n ⁑ ( x ) ⁒ x = H ^ n + 1 ⁑ ( x ) ⁒ x + n 2 ⁒ H ^ n - 1 ⁑ ( x ) ⁒ x π‘₯ Hermite-polynomial-monic 𝑛 π‘₯ π‘₯ Hermite-polynomial-monic 𝑛 1 π‘₯ π‘₯ 𝑛 2 Hermite-polynomial-monic 𝑛 1 π‘₯ π‘₯ {\displaystyle{\displaystyle{\displaystyle x{\widehat{H}}_{n}\left(x\right){x}% ={\widehat{H}}_{n+1}\left(x\right){x}+\frac{n}{2}{\widehat{H}}_{n-1}\left(x% \right){x}}}} {\displaystyle x\monicHermite{n}@@{x}{x}=\monicHermite{n+1}@@{x}{x}+\frac{n}{2}\monicHermite{n-1}@@{x}{x} }
H n ⁑ ( x ) = 2 n ⁒ H ^ n ⁑ ( x ) ⁒ x Hermite-polynomial-H 𝑛 π‘₯ superscript 2 𝑛 Hermite-polynomial-monic 𝑛 π‘₯ π‘₯ {\displaystyle{\displaystyle{\displaystyle H_{n}\left(x\right)=2^{n}{\widehat{% H}}_{n}\left(x\right){x}}}} {\displaystyle \Hermite{n}@{x}=2^n\monicHermite{n}@@{x}{x} }

Differential equation

y β€²β€² ⁒ ( x ) - 2 ⁒ x ⁒ y β€² ⁒ ( x ) + 2 ⁒ n ⁒ y ⁒ ( x ) = 0 superscript 𝑦 β€²β€² π‘₯ 2 π‘₯ superscript 𝑦 β€² π‘₯ 2 𝑛 𝑦 π‘₯ 0 {\displaystyle{\displaystyle{\displaystyle y^{\prime\prime}(x)-2xy^{\prime}(x)% +2ny(x)=0}}} {\displaystyle y''(x)-2xy'(x)+2ny(x)=0 }

Substitution(s): y ⁒ ( x ) = H n ⁑ ( x ) 𝑦 π‘₯ Hermite-polynomial-H 𝑛 π‘₯ {\displaystyle{\displaystyle{\displaystyle y(x)=H_{n}\left(x\right)}}}


Forward shift operator

d d ⁒ x ⁒ H n ⁑ ( x ) = 2 ⁒ n ⁒ H n - 1 ⁑ ( x ) 𝑑 𝑑 π‘₯ Hermite-polynomial-H 𝑛 π‘₯ 2 𝑛 Hermite-polynomial-H 𝑛 1 π‘₯ {\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}H_{n}\left(x\right)=2nH_% {n-1}\left(x\right)}}} {\displaystyle \frac{d}{dx}\Hermite{n}@{x}=2n\Hermite{n-1}@{x} }

Backward shift operator

d d ⁒ x ⁒ H n ⁑ ( x ) - 2 ⁒ x ⁒ H n ⁑ ( x ) = - H n + 1 ⁑ ( x ) 𝑑 𝑑 π‘₯ Hermite-polynomial-H 𝑛 π‘₯ 2 π‘₯ Hermite-polynomial-H 𝑛 π‘₯ Hermite-polynomial-H 𝑛 1 π‘₯ {\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}H_{n}\left(x\right)-2xH_% {n}\left(x\right)=-H_{n+1}\left(x\right)}}} {\displaystyle \frac{d}{dx}\Hermite{n}@{x}-2x\Hermite{n}@{x}=-\Hermite{n+1}@{x} }
d d ⁒ x ⁒ [ e - x 2 ⁒ H n ⁑ ( x ) ] = - e - x 2 ⁒ H n + 1 ⁑ ( x ) 𝑑 𝑑 π‘₯ delimited-[] superscript π‘₯ 2 Hermite-polynomial-H 𝑛 π‘₯ superscript π‘₯ 2 Hermite-polynomial-H 𝑛 1 π‘₯ {\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left[{\mathrm{e}^{-x^{2% }}}H_{n}\left(x\right)\right]=-{\mathrm{e}^{-x^{2}}}H_{n+1}\left(x\right)}}} {\displaystyle \frac{d}{dx}\left[\expe^{-x^2}\Hermite{n}@{x}\right]=-\expe^{-x^2}\Hermite{n+1}@{x} }

Rodrigues-type formula

e - x 2 ⁒ H n ⁑ ( x ) = ( - 1 ) n ⁒ ( d d ⁒ x ) n ⁒ [ e - x 2 ] superscript π‘₯ 2 Hermite-polynomial-H 𝑛 π‘₯ superscript 1 𝑛 superscript 𝑑 𝑑 π‘₯ 𝑛 delimited-[] superscript π‘₯ 2 {\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{-x^{2}}}H_{n}\left(x% \right)=(-1)^{n}\left(\frac{d}{dx}\right)^{n}\left[{\mathrm{e}^{-x^{2}}}\right% ]}}} {\displaystyle \expe^{-x^2}\Hermite{n}@{x}=(-1)^n\left(\frac{d}{dx}\right)^n\left[\expe^{-x^2}\right] }

Generating functions

exp ⁑ ( 2 ⁒ x ⁒ t - t 2 ) = βˆ‘ n = 0 ∞ H n ⁑ ( x ) n ! ⁒ t n 2 π‘₯ 𝑑 superscript 𝑑 2 superscript subscript 𝑛 0 Hermite-polynomial-H 𝑛 π‘₯ 𝑛 superscript 𝑑 𝑛 {\displaystyle{\displaystyle{\displaystyle\exp\left(2xt-t^{2}\right)=\sum_{n=0% }^{\infty}\frac{H_{n}\left(x\right)}{n!}t^{n}}}} {\displaystyle \exp@{2xt-t^2}=\sum_{n=0}^{\infty}\frac{\Hermite{n}@{x}}{n!}t^n }
e t ⁒ cos ⁑ ( 2 ⁒ x ⁒ t ) = βˆ‘ n = 0 ∞ ( - 1 ) n ( 2 ⁒ n ) ! ⁒ H 2 ⁒ n ⁑ ( x ) ⁒ t n 𝑑 2 π‘₯ 𝑑 superscript subscript 𝑛 0 superscript 1 𝑛 2 𝑛 Hermite-polynomial-H 2 𝑛 π‘₯ superscript 𝑑 𝑛 {\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{t}}\cos\left(2x\sqrt{t}% \right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n)!}H_{2n}\left(x\right)t^{n}}}} {\displaystyle \expe^t\cos@{2x\sqrt{t}}=\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}\Hermite{2n}@{x}t^n }
e t t ⁒ sin ⁑ ( 2 ⁒ x ⁒ t ) = βˆ‘ n = 0 ∞ ( - 1 ) n ( 2 ⁒ n + 1 ) ! ⁒ H 2 ⁒ n + 1 ⁑ ( x ) ⁒ t n 𝑑 𝑑 2 π‘₯ 𝑑 superscript subscript 𝑛 0 superscript 1 𝑛 2 𝑛 1 Hermite-polynomial-H 2 𝑛 1 π‘₯ superscript 𝑑 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{{\mathrm{e}^{t}}}{\sqrt{t}}% \sin\left(2x\sqrt{t}\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)!}H_{2n+1% }\left(x\right)t^{n}}}} {\displaystyle \frac{\expe^t}{\sqrt{t}}\sin@{2x\sqrt{t}}=\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}\Hermite{2n+1}@{x}t^n }
e - t 2 ⁒ cosh ⁑ ( 2 ⁒ x ⁒ t ) = βˆ‘ n = 0 ∞ H 2 ⁒ n ⁑ ( x ) ( 2 ⁒ n ) ! ⁒ t 2 ⁒ n superscript 𝑑 2 2 π‘₯ 𝑑 superscript subscript 𝑛 0 Hermite-polynomial-H 2 𝑛 π‘₯ 2 𝑛 superscript 𝑑 2 𝑛 {\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{-t^{2}}}\cosh\left(2xt% \right)=\sum_{n=0}^{\infty}\frac{H_{2n}\left(x\right)}{(2n)!}t^{2n}}}} {\displaystyle \expe^{-t^2}\cosh@{2xt}=\sum_{n=0}^{\infty} \frac{\Hermite{2n}@{x}}{(2n)!}t^{2n} }
e - t 2 ⁒ sinh ⁑ ( 2 ⁒ x ⁒ t ) = βˆ‘ n = 0 ∞ H 2 ⁒ n + 1 ⁑ ( x ) ( 2 ⁒ n + 1 ) ! ⁒ t 2 ⁒ n + 1 superscript 𝑑 2 2 π‘₯ 𝑑 superscript subscript 𝑛 0 Hermite-polynomial-H 2 𝑛 1 π‘₯ 2 𝑛 1 superscript 𝑑 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{-t^{2}}}\sinh\left(2xt% \right)=\sum_{n=0}^{\infty}\frac{H_{2n+1}\left(x\right)}{(2n+1)!}t^{2n+1}}}} {\displaystyle \expe^{-t^2}\sinh@{2xt}=\sum_{n=0}^{\infty} \frac{\Hermite{2n+1}@{x}}{(2n+1)!}t^{2n+1} }
( 1 + t 2 ) - Ξ³ ⁒ \HyperpFq ⁒ 11 ⁒ @ ⁒ @ ⁒ Ξ³ ⁒ 1 2 ⁒ x 2 ⁒ t 2 1 + t 2 = βˆ‘ n = 0 ∞ ( Ξ³ ) n ( 2 ⁒ n ) ! ⁒ H 2 ⁒ n ⁑ ( x ) ⁒ t 2 ⁒ n superscript 1 superscript 𝑑 2 𝛾 \HyperpFq 11 @ @ 𝛾 1 2 superscript π‘₯ 2 superscript 𝑑 2 1 superscript 𝑑 2 superscript subscript 𝑛 0 Pochhammer-symbol 𝛾 𝑛 2 𝑛 Hermite-polynomial-H 2 𝑛 π‘₯ superscript 𝑑 2 𝑛 {\displaystyle{\displaystyle{\displaystyle(1+t^{2})^{-\gamma}\,\HyperpFq{1}{1}% @@{\gamma}{\frac{1}{2}}{\frac{x^{2}t^{2}}{1+t^{2}}}=\sum_{n=0}^{\infty}\frac{{% \left(\gamma\right)_{n}}}{(2n)!}H_{2n}\left(x\right)t^{2n}}}} {\displaystyle (1+t^2)^{-\gamma}\,\HyperpFq{1}{1}@@{\gamma}{\frac{1}{2}}{\frac{x^2t^2}{1+t^2}}= \sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}}{(2n)!}\Hermite{2n}@{x}t^{2n} }
x ⁒ t 1 + t 2 ⁒ \HyperpFq ⁒ 11 ⁒ @ ⁒ @ ⁒ Ξ³ + 1 2 ⁒ 3 2 ⁒ x 2 ⁒ t 2 1 + t 2 = βˆ‘ n = 0 ∞ ( Ξ³ + 1 2 ) n ( 2 ⁒ n + 1 ) ! ⁒ H 2 ⁒ n + 1 ⁑ ( x ) ⁒ t 2 ⁒ n + 1 π‘₯ 𝑑 1 superscript 𝑑 2 \HyperpFq 11 @ @ 𝛾 1 2 3 2 superscript π‘₯ 2 superscript 𝑑 2 1 superscript 𝑑 2 superscript subscript 𝑛 0 Pochhammer-symbol 𝛾 1 2 𝑛 2 𝑛 1 Hermite-polynomial-H 2 𝑛 1 π‘₯ superscript 𝑑 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\frac{xt}{\sqrt{1+t^{2}}}\ \HyperpFq% {1}{1}@@{\gamma+\frac{1}{2}}{\frac{3}{2}}{\frac{x^{2}t^{2}}{1+t^{2}}}=\sum_{n=% 0}^{\infty}\frac{{\left(\gamma+\frac{1}{2}\right)_{n}}}{(2n+1)!}H_{2n+1}\left(% x\right)t^{2n+1}}}} {\displaystyle \frac{xt}{\sqrt{1+t^2}}\ \HyperpFq{1}{1}@@{\gamma+\frac{1}{2}}{\frac{3}{2}}{\frac{x^2t^2}{1+t^2}} =\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma+\frac{1}{2}}{n}}{(2n+1)!}\Hermite{2n+1}@{x}t^{2n+1} }
1 + 2 ⁒ x ⁒ t + 4 ⁒ t 2 ( 1 + 4 ⁒ t 2 ) 3 2 ⁒ exp ⁑ ( 4 ⁒ x 2 ⁒ t 2 1 + 4 ⁒ t 2 ) = βˆ‘ n = 0 ∞ H n ⁑ ( x ) ⌊ n / 2 βŒ‹ ! ⁒ t n 1 2 π‘₯ 𝑑 4 superscript 𝑑 2 superscript 1 4 superscript 𝑑 2 3 2 4 superscript π‘₯ 2 superscript 𝑑 2 1 4 superscript 𝑑 2 superscript subscript 𝑛 0 Hermite-polynomial-H 𝑛 π‘₯ 𝑛 2 superscript 𝑑 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1+2xt+4t^{2}}{(1+4t^{2})^{% \frac{3}{2}}}\exp\left(\frac{4x^{2}t^{2}}{1+4t^{2}}\right)=\sum_{n=0}^{\infty}% \frac{H_{n}\left(x\right)}{\lfloor n/2\rfloor\,!}t^{n}}}} {\displaystyle \frac{1+2xt+4t^2}{(1+4t^2)^{\frac{3}{2}}}\exp@{\frac{4x^2t^2}{1+4t^2}} =\sum_{n=0}^{\infty}\frac{\Hermite{n}@{x}}{\lfloor n/2\rfloor\,!}t^n }

Limit relations

Meixner-Pollaczek polynomial to Hermite polynomial

lim Ξ» β†’ ∞ ⁑ Ξ» - 1 2 ⁒ n ⁒ P n ( Ξ» ) ⁑ ( ( sin ⁑ Ο• ) - 1 ⁒ ( x ⁒ Ξ» - Ξ» ⁒ cos ⁑ Ο• ) ; Ο• ) = H n ⁑ ( x ) n ! subscript β†’ πœ† superscript πœ† 1 2 𝑛 Meixner-Pollaczek-polynomial-P πœ† 𝑛 superscript italic-Ο• 1 π‘₯ πœ† πœ† italic-Ο• italic-Ο• Hermite-polynomial-H 𝑛 π‘₯ 𝑛 {\displaystyle{\displaystyle{\displaystyle\lim_{\lambda\rightarrow\infty}% \lambda^{-\frac{1}{2}n}P^{(\lambda)}_{n}\!\left((\sin\phi)^{-1}(x\sqrt{\lambda% }-\lambda\cos\phi);\phi\right)=\frac{H_{n}\left(x\right)}{n!}}}} {\displaystyle \lim_{\lambda\rightarrow\infty} \lambda^{-\frac{1}{2}n}\MeixnerPollaczek{\lambda}{n}@{(\sin@@{\phi})^{-1}(x\sqrt{\lambda}-\lambda\cos@@{\phi})}{\phi}=\frac{\Hermite{n}@{x}}{n!} }

Jacobi polynomial to Hermite polynomial

lim Ξ± β†’ ∞ ⁑ Ξ± - 1 2 ⁒ n ⁒ P n ( Ξ± , Ξ± ) ⁑ ( Ξ± - 1 2 ⁒ x ) = H n ⁑ ( x ) 2 n ⁒ n ! subscript β†’ 𝛼 superscript 𝛼 1 2 𝑛 Jacobi-polynomial-P 𝛼 𝛼 𝑛 superscript 𝛼 1 2 π‘₯ Hermite-polynomial-H 𝑛 π‘₯ superscript 2 𝑛 𝑛 {\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}\alpha% ^{-\frac{1}{2}n}P^{(\alpha,\alpha)}_{n}\left(\alpha^{-\frac{1}{2}}x\right)=% \frac{H_{n}\left(x\right)}{2^{n}n!}}}} {\displaystyle \lim_{\alpha\rightarrow\infty} \alpha^{-\frac{1}{2}n}\Jacobi{\alpha}{\alpha}{n}@{\alpha^{-\frac{1}{2}}x}=\frac{\Hermite{n}@{x}}{2^nn!} }

Gegenbauer / Ultraspherical polynomial to Hermite polynomial

lim Ξ± β†’ ∞ ⁑ Ξ± - 1 2 ⁒ n ⁒ C n Ξ± + 1 2 ⁑ ( Ξ± - 1 2 ⁒ x ) = H n ⁑ ( x ) n ! subscript β†’ 𝛼 superscript 𝛼 1 2 𝑛 ultraspherical-Gegenbauer-polynomial 𝛼 1 2 𝑛 superscript 𝛼 1 2 π‘₯ Hermite-polynomial-H 𝑛 π‘₯ 𝑛 {\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}\alpha% ^{-\frac{1}{2}n}C^{\alpha+\frac{1}{2}}_{n}\left(\alpha^{-\frac{1}{2}}x\right)=% \frac{H_{n}\left(x\right)}{n!}}}} {\displaystyle \lim_{\alpha\rightarrow\infty} \alpha^{-\frac{1}{2}n}\Ultra{\alpha+\frac{1}{2}}{n}@{\alpha^{-\frac{1}{2}}x}=\frac{\Hermite{n}@{x}}{n!} }

Krawtchouk polynomial to Hermite polynomial

lim N β†’ ∞ ⁑ \binomial ⁒ N ⁒ n ⁒ K n ⁑ ( p ⁒ N + x ⁒ 2 ⁒ p ⁒ ( 1 - p ) ⁒ N ; p , N ) = ( - 1 ) n ⁒ H n ⁑ ( x ) 2 n ⁒ n ! ⁒ ( p 1 - p ) n subscript β†’ 𝑁 \binomial 𝑁 𝑛 Krawtchouk-polynomial-K 𝑛 𝑝 𝑁 π‘₯ 2 𝑝 1 𝑝 𝑁 𝑝 𝑁 superscript 1 𝑛 Hermite-polynomial-H 𝑛 π‘₯ superscript 2 𝑛 𝑛 superscript 𝑝 1 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}\sqrt{% \binomial{N}{n}}K_{n}\!\left(pN+x\sqrt{2p(1-p)N};p,N\right)=\frac{% \displaystyle(-1)^{n}H_{n}\left(x\right)}{\displaystyle\sqrt{2^{n}n!\left(% \frac{p}{1-p}\right)^{n}}}}}} {\displaystyle \lim_{N\rightarrow\infty} \sqrt{\binomial{N}{n}}\Krawtchouk{n}@{pN+x\sqrt{2p(1-p)N}}{p}{N} =\frac{\displaystyle (-1)^n\Hermite{n}@{x}}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}} }

Laguerre polynomial to Hermite polynomial

lim Ξ± β†’ ∞ ( 2 Ξ± ) 1 2 ⁒ n L n Ξ± ⁑ ( ( 2 Ξ± ) 1 2 x + Ξ± ) = ( - 1 ) n n ! H n ⁑ ( x ) fragments subscript β†’ 𝛼 superscript fragments ( 2 𝛼 ) 1 2 𝑛 superscript generalized-Laguerre-polynomial-L 𝛼 𝑛 fragments ( 2 Ξ± 1 2 x Ξ± ) superscript 1 𝑛 𝑛 Hermite-polynomial-H 𝑛 π‘₯ {\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} }

Charlier polynomial to Hermite polynomial

lim a β†’ ∞ ⁑ ( 2 ⁒ a ) 1 2 ⁒ n ⁒ C n ⁑ ( ( 2 ⁒ a ) 1 2 ⁒ x + a ; a ) = ( - 1 ) n ⁒ H n ⁑ ( x ) subscript β†’ π‘Ž superscript 2 π‘Ž 1 2 𝑛 Charlier-polynomial-C 𝑛 superscript 2 π‘Ž 1 2 π‘₯ π‘Ž π‘Ž superscript 1 𝑛 Hermite-polynomial-H 𝑛 π‘₯ {\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow\infty}(2a)^{\frac% {1}{2}n}C_{n}\!\left((2a)^{\frac{1}{2}}x+a;a\right)=(-1)^{n}H_{n}\left(x\right% )}}} {\displaystyle \lim_{a\rightarrow\infty} (2a)^{\frac{1}{2}n}\Charlier{n}@{(2a)^{\frac{1}{2}}x+a}{a}=(-1)^n\Hermite{n}@{x} }

Remarks

H n ⁑ ( x ) n ! = βˆ‘ k = 0 ⌊ n / 2 βŒ‹ ( - 1 ) k ⁒ ( 2 ⁒ x ) n - 2 ⁒ k k ! ⁒ ( n - 2 ⁒ k ) ! Hermite-polynomial-H 𝑛 π‘₯ 𝑛 superscript subscript π‘˜ 0 𝑛 2 superscript 1 π‘˜ superscript 2 π‘₯ 𝑛 2 π‘˜ π‘˜ 𝑛 2 π‘˜ {\displaystyle{\displaystyle{\displaystyle\frac{H_{n}\left(x\right)}{n!}=\sum_% {k=0}^{\lfloor n/2\rfloor}\frac{(-1)^{k}(2x)^{n-2k}}{k!\,(n-2k)!}}}} {\displaystyle \frac{\Hermite{n}@{x}}{n!}=\sum_{k=0}^{\lfloor n/2\rfloor} \frac{(-1)^k(2x)^{n-2k}}{k!\,(n-2k)!} }
H 2 ⁒ n ⁑ ( x ) = ( - 1 ) n ⁒ n ! ⁒  2 2 ⁒ n ⁒ L n - 1 2 ⁑ ( x 2 ) Hermite-polynomial-H 2 𝑛 π‘₯ superscript 1 𝑛 𝑛 superscript  2 2 𝑛 generalized-Laguerre-polynomial-L 1 2 𝑛 superscript π‘₯ 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 π‘₯ superscript 1 𝑛 𝑛 superscript  2 2 𝑛 1 π‘₯ generalized-Laguerre-polynomial-L 1 2 𝑛 superscript π‘₯ 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} }
1 2 ⁒ Ο€ ⁒ ∫ - ∞ ∞ H n ⁑ ( y ) ⁒ e - 1 2 ⁒ y 2 ⁒ e i ⁒ x ⁒ y ⁒ 𝑑 y = i n ⁒ H n ⁑ ( x ) ⁒ e - 1 2 ⁒ x 2 1 2 superscript subscript Hermite-polynomial-H 𝑛 𝑦 1 2 superscript 𝑦 2 imaginary-unit π‘₯ 𝑦 differential-d 𝑦 imaginary-unit 𝑛 Hermite-polynomial-H 𝑛 π‘₯ 1 2 superscript π‘₯ 2 {\displaystyle{\displaystyle{\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^% {\infty}H_{n}\left(y\right){\mathrm{e}^{-\frac{1}{2}y^{2}}}{\mathrm{e}^{% \mathrm{i}xy}}dy={\mathrm{i}^{n}}H_{n}\left(x\right){\mathrm{e}^{-\frac{1}{2}x% ^{2}}}}}} {\displaystyle \frac1{\sqrt{2\cpi}} \int_{-\infty}^\infty \Hermite{n}@{y} \expe^{-\frac12 y^2} \expe^{\iunit xy} dy= \iunit^n \Hermite{n}@{x} \expe^{-\frac12 x^2} }
1 Ο€ ⁒ ∫ - ∞ ∞ H n ⁑ ( y ) ⁒ e - y 2 ⁒ e i ⁒ x ⁒ y ⁒ 𝑑 y = i n ⁒ x n ⁒ e - 1 4 ⁒ x 2 1 superscript subscript Hermite-polynomial-H 𝑛 𝑦 superscript 𝑦 2 imaginary-unit π‘₯ 𝑦 differential-d 𝑦 imaginary-unit 𝑛 superscript π‘₯ 𝑛 1 4 superscript π‘₯ 2 {\displaystyle{\displaystyle{\displaystyle\frac{1}{\sqrt{\pi}}\int_{-\infty}^{% \infty}H_{n}\left(y\right){\mathrm{e}^{-y^{2}}}{\mathrm{e}^{\mathrm{i}xy}}dy={% \mathrm{i}^{n}}x^{n}{\mathrm{e}^{-\frac{1}{4}x^{2}}}}}} {\displaystyle \frac1{\sqrt\cpi} \int_{-\infty}^\infty \Hermite{n}@{y} \expe^{-y^2} \expe^{\iunit xy} dy= \iunit^n x^n \expe^{-\frac14 x^2} }
i n 2 ⁒ Ο€ ⁒ ∫ - ∞ ∞ y n ⁒ e - 1 4 ⁒ y 2 ⁒ e - i ⁒ x ⁒ y ⁒ 𝑑 y = H n ⁑ ( x ) ⁒ e - x 2 imaginary-unit 𝑛 2 superscript subscript superscript 𝑦 𝑛 1 4 superscript 𝑦 2 imaginary-unit π‘₯ 𝑦 differential-d 𝑦 Hermite-polynomial-H 𝑛 π‘₯ superscript π‘₯ 2 {\displaystyle{\displaystyle{\displaystyle\frac{{\mathrm{i}^{n}}}{2\sqrt{\pi}}% \int_{-\infty}^{\infty}y^{n}{\mathrm{e}^{-\frac{1}{4}y^{2}}}{\mathrm{e}^{-% \mathrm{i}xy}}dy=H_{n}\left(x\right){\mathrm{e}^{-x^{2}}}}}} {\displaystyle \frac{\iunit^n}{2\sqrt\cpi} \int_{-\infty}^\infty y^n \expe^{-\frac14 y^2} \expe^{-\iunit xy} dy= \Hermite{n}@{x} \expe^{-x^2} }

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