H n β‘ ( x ) = ( 2 x ) n \HyperpFq 20 @ @ - n / 2 , - ( n - 1 ) / 2 - - 1 x 2 fragments Hermite-polynomial-H π π₯ superscript fragments ( 2 x ) π \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}} }
1 Ο β’ β« - β β e - x 2 β’ H m β‘ ( x ) β’ H n β‘ ( x ) β’ π x = 2 n β’ n ! β’ Ξ΄ m , n 1 superscript subscript superscript π₯ 2 Hermite-polynomial-H π π₯ Hermite-polynomial-H π π₯ differential-d π₯ superscript 2 π π Kronecker-delta π π {\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} }
H n + 1 β‘ ( x ) - 2 β’ x β’ H n β‘ ( x ) + 2 β’ n β’ H n - 1 β‘ ( x ) = 0 Hermite-polynomial-H π 1 π₯ 2 π₯ Hermite-polynomial-H π π₯ 2 π Hermite-polynomial-H π 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 }
x β’ H ^ n β‘ ( x ) β’ x = H ^ n + 1 β‘ ( x ) β’ x + n 2 β’ H ^ n - 1 β‘ ( x ) β’ x π₯ Hermite-polynomial-monic π π₯ π₯ Hermite-polynomial-monic π 1 π₯ π₯ π 2 Hermite-polynomial-monic π 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 β‘ ( x ) = 2 n β’ H ^ n β‘ ( x ) β’ x Hermite-polynomial-H π π₯ superscript 2 π Hermite-polynomial-monic π 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} }
y β²β² β’ ( x ) - 2 β’ x β’ y β² β’ ( x ) + 2 β’ n β’ y β’ ( x ) = 0 superscript π¦ β²β² π₯ 2 π₯ superscript π¦ β² π₯ 2 π π¦ π₯ 0 {\displaystyle{\displaystyle{\displaystyle y^{\prime\prime}(x)-2xy^{\prime}(x)% +2ny(x)=0}}} {\displaystyle y''(x)-2xy'(x)+2ny(x)=0 }
d d β’ x β’ H n β‘ ( x ) = 2 β’ n β’ H n - 1 β‘ ( x ) π π π₯ Hermite-polynomial-H π π₯ 2 π Hermite-polynomial-H π 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} }
d d β’ x β’ H n β‘ ( x ) - 2 β’ x β’ H n β‘ ( x ) = - H n + 1 β‘ ( x ) π π π₯ Hermite-polynomial-H π π₯ 2 π₯ Hermite-polynomial-H π π₯ Hermite-polynomial-H π 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 β’ x β’ [ e - x 2 β’ H n β‘ ( x ) ] = - e - x 2 β’ H n + 1 β‘ ( x ) π π π₯ delimited-[] superscript π₯ 2 Hermite-polynomial-H π π₯ superscript π₯ 2 Hermite-polynomial-H π 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} }
e - x 2 β’ H n β‘ ( x ) = ( - 1 ) n β’ ( d d β’ x ) n β’ [ e - x 2 ] superscript π₯ 2 Hermite-polynomial-H π π₯ superscript 1 π superscript π π π₯ π 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] }
exp β‘ ( 2 β’ x β’ t - t 2 ) = β n = 0 β H n β‘ ( x ) n ! β’ t n 2 π₯ π‘ superscript π‘ 2 superscript subscript π 0 Hermite-polynomial-H π π₯ π superscript π‘ π {\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 β’ cos β‘ ( 2 β’ x β’ t ) = β n = 0 β ( - 1 ) n ( 2 β’ n ) ! β’ H 2 β’ n β‘ ( x ) β’ t n π‘ 2 π₯ π‘ superscript subscript π 0 superscript 1 π 2 π Hermite-polynomial-H 2 π π₯ superscript π‘ π {\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 β’ sin β‘ ( 2 β’ x β’ t ) = β n = 0 β ( - 1 ) n ( 2 β’ n + 1 ) ! β’ H 2 β’ n + 1 β‘ ( x ) β’ t n π‘ π‘ 2 π₯ π‘ superscript subscript π 0 superscript 1 π 2 π 1 Hermite-polynomial-H 2 π 1 π₯ superscript π‘ π {\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 β’ cosh β‘ ( 2 β’ x β’ t ) = β n = 0 β H 2 β’ n β‘ ( x ) ( 2 β’ n ) ! β’ t 2 β’ n superscript π‘ 2 2 π₯ π‘ superscript subscript π 0 Hermite-polynomial-H 2 π π₯ 2 π superscript π‘ 2 π {\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 β’ sinh β‘ ( 2 β’ x β’ t ) = β n = 0 β H 2 β’ n + 1 β‘ ( x ) ( 2 β’ n + 1 ) ! β’ t 2 β’ n + 1 superscript π‘ 2 2 π₯ π‘ superscript subscript π 0 Hermite-polynomial-H 2 π 1 π₯ 2 π 1 superscript π‘ 2 π 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 β’ \HyperpFq β’ 11 β’ @ β’ @ β’ Ξ³ β’ 1 2 β’ x 2 β’ t 2 1 + t 2 = β n = 0 β ( Ξ³ ) n ( 2 β’ n ) ! β’ H 2 β’ n β‘ ( x ) β’ t 2 β’ n superscript 1 superscript π‘ 2 πΎ \HyperpFq 11 @ @ πΎ 1 2 superscript π₯ 2 superscript π‘ 2 1 superscript π‘ 2 superscript subscript π 0 Pochhammer-symbol πΎ π 2 π Hermite-polynomial-H 2 π π₯ superscript π‘ 2 π {\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 β’ t 1 + t 2 β’ \HyperpFq β’ 11 β’ @ β’ @ β’ Ξ³ + 1 2 β’ 3 2 β’ x 2 β’ t 2 1 + t 2 = β n = 0 β ( Ξ³ + 1 2 ) n ( 2 β’ n + 1 ) ! β’ H 2 β’ n + 1 β‘ ( x ) β’ t 2 β’ n + 1 π₯ π‘ 1 superscript π‘ 2 \HyperpFq 11 @ @ πΎ 1 2 3 2 superscript π₯ 2 superscript π‘ 2 1 superscript π‘ 2 superscript subscript π 0 Pochhammer-symbol πΎ 1 2 π 2 π 1 Hermite-polynomial-H 2 π 1 π₯ superscript π‘ 2 π 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 β’ x β’ t + 4 β’ t 2 ( 1 + 4 β’ t 2 ) 3 2 β’ exp β‘ ( 4 β’ x 2 β’ t 2 1 + 4 β’ t 2 ) = β n = 0 β H n β‘ ( x ) β n / 2 β ! β’ t n 1 2 π₯ π‘ 4 superscript π‘ 2 superscript 1 4 superscript π‘ 2 3 2 4 superscript π₯ 2 superscript π‘ 2 1 4 superscript π‘ 2 superscript subscript π 0 Hermite-polynomial-H π π₯ π 2 superscript π‘ π {\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 }
lim Ξ» β β β‘ Ξ» - 1 2 β’ n β’ \MeixnerPollaczek β’ Ξ» β’ n β’ @ β’ ( sin β‘ Ο ) - 1 β’ ( x β’ Ξ» - Ξ» β’ cos β‘ Ο ) β’ Ο = \Hermite β’ n β’ @ β’ x n ! subscript β π superscript π 1 2 π \MeixnerPollaczek π π @ superscript italic-Ο 1 π₯ π π italic-Ο italic-Ο \Hermite π @ π₯ π {\displaystyle{\displaystyle{\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!}}}} {\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!} }
lim Ξ± β β β‘ Ξ± - 1 2 β’ n β’ P n ( Ξ± , Ξ± ) β‘ ( Ξ± - 1 2 β’ x ) = H n β‘ ( x ) 2 n β’ n ! subscript β πΌ superscript πΌ 1 2 π Jacobi-polynomial-P πΌ πΌ π superscript πΌ 1 2 π₯ Hermite-polynomial-H π π₯ superscript 2 π π {\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!} }
lim Ξ± β β β‘ Ξ± - 1 2 β’ n β’ C n Ξ± + 1 2 β‘ ( Ξ± - 1 2 β’ x ) = H n β‘ ( x ) n ! subscript β πΌ superscript πΌ 1 2 π ultraspherical-Gegenbauer-polynomial πΌ 1 2 π superscript πΌ 1 2 π₯ Hermite-polynomial-H π π₯ π {\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!} }
lim N β β β‘ \binomial β’ N β’ n β’ K n β‘ ( p β’ N + x β’ 2 β’ p β’ ( 1 - p ) β’ N ; p , N ) = ( - 1 ) n β’ H n β‘ ( x ) 2 n β’ n ! β’ ( p 1 - p ) n subscript β π \binomial π π Krawtchouk-polynomial-K π π π π₯ 2 π 1 π π π π superscript 1 π Hermite-polynomial-H π π₯ superscript 2 π π superscript π 1 π π {\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}} }
lim Ξ± β β ( 2 Ξ± ) 1 2 β’ n L n Ξ± β‘ ( ( 2 Ξ± ) 1 2 x + Ξ± ) = ( - 1 ) n n ! H n β‘ ( x ) fragments subscript β πΌ superscript fragments ( 2 πΌ ) 1 2 π superscript generalized-Laguerre-polynomial-L πΌ π fragments ( 2 Ξ± 1 2 x Ξ± ) superscript 1 π π Hermite-polynomial-H π 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} }
lim a β β β‘ ( 2 β’ a ) 1 2 β’ n β’ C n β‘ ( ( 2 β’ a ) 1 2 β’ x + a ; a ) = ( - 1 ) n β’ H n β‘ ( x ) subscript β π superscript 2 π 1 2 π Charlier-polynomial-C π superscript 2 π 1 2 π₯ π π superscript 1 π Hermite-polynomial-H π 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} }
H n β‘ ( x ) n ! = β k = 0 β n / 2 β ( - 1 ) k β’ ( 2 β’ x ) n - 2 β’ k k ! β’ ( n - 2 β’ k ) ! Hermite-polynomial-H π π₯ π superscript subscript π 0 π 2 superscript 1 π superscript 2 π₯ π 2 π π π 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 β’ n β‘ ( x ) = ( - 1 ) n β’ n ! β’ β2 2 β’ n β’ L n - 1 2 β‘ ( x 2 ) Hermite-polynomial-H 2 π π₯ superscript 1 π π superscript β2 2 π generalized-Laguerre-polynomial-L 1 2 π 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 β’ n + 1 β‘ ( x ) = ( - 1 ) n β’ n ! β’ β2 2 β’ n + 1 β’ x β’ L n 1 2 β‘ ( x 2 ) Hermite-polynomial-H 2 π 1 π₯ superscript 1 π π superscript β2 2 π 1 π₯ generalized-Laguerre-polynomial-L 1 2 π 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 β’ Ο β’ β« - β β H n β‘ ( y ) β’ e - 1 2 β’ y 2 β’ e i β’ x β’ y β’ π y = i n β’ H n β‘ ( x ) β’ e - 1 2 β’ x 2 1 2 superscript subscript Hermite-polynomial-H π π¦ 1 2 superscript π¦ 2 imaginary-unit π₯ π¦ differential-d π¦ imaginary-unit π Hermite-polynomial-H π 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 β’ β« - β β H n β‘ ( y ) β’ e - y 2 β’ e i β’ x β’ y β’ π y = i n β’ x n β’ e - 1 4 β’ x 2 1 superscript subscript Hermite-polynomial-H π π¦ superscript π¦ 2 imaginary-unit π₯ π¦ differential-d π¦ imaginary-unit π superscript π₯ π 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 β’ Ο β’ β« - β β y n β’ e - 1 4 β’ y 2 β’ e - i β’ x β’ y β’ π y = H n β‘ ( x ) β’ e - x 2 imaginary-unit π 2 superscript subscript superscript π¦ π 1 4 superscript π¦ 2 imaginary-unit π₯ π¦ differential-d π¦ Hermite-polynomial-H π 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} }