Hermite

<< Charlier

Hermite

Orthogonal Polynomials >>

Hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle H_{n}\left(x\right)=(2x)^{n}\,\HyperpFq{2}{0}@@{-n/2,-(n-1)/2}{-}{-\frac{1}{x^{2}}}}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}{\mathrm{e}^{-x^{2}}}H_{m}\left(x\right)H_{n}\left(x\right)\,dx=2^{n}n!\,\delta_{m,n}}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle H_{n+1}\left(x\right)-2xH_{n}\left(x\right)+2nH_{n-1}\left(x\right)=0}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{H}}_{n}\left(x\right){x}={\widehat{H}}_{n+1}\left(x\right){x}+\frac{n}{2}{\widehat{H}}_{n-1}\left(x\right){x}}}}$
${\displaystyle{\displaystyle{\displaystyle H_{n}\left(x\right)=2^{n}{\widehat{H}}_{n}\left(x\right){x}}}}$

Differential equation

${\displaystyle{\displaystyle{\displaystyle y^{\prime\prime}(x)-2xy^{\prime}(x)+2ny(x)=0}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle y(x)=H_{n}\left(x\right)}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}H_{n}\left(x\right)=2nH_{n-1}\left(x\right)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}H_{n}\left(x\right)-2xH_{n}\left(x\right)=-H_{n+1}\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left[{\mathrm{e}^{-x^{2}}}H_{n}\left(x\right)\right]=-{\mathrm{e}^{-x^{2}}}H_{n+1}\left(x\right)}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{-x^{2}}}H_{n}\left(x\right)=(-1)^{n}\left(\frac{d}{dx}\right)^{n}\left[{\mathrm{e}^{-x^{2}}}\right]}}}$

Generating functions

${\displaystyle{\displaystyle{\displaystyle\exp\left(2xt-t^{2}\right)=\sum_{n=0}^{\infty}\frac{H_{n}\left(x\right)}{n!}t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{t}}\cos\left(2x\sqrt{t}\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n)!}H_{2n}\left(x\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{{\mathrm{e}^{t}}}{\sqrt{t}}\sin\left(2x\sqrt{t}\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)!}H_{2n+1}\left(x\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{-t^{2}}}\cosh\left(2xt\right)=\sum_{n=0}^{\infty}\frac{H_{2n}\left(x\right)}{(2n)!}t^{2n}}}}$
${\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{-t^{2}}}\sinh\left(2xt\right)=\sum_{n=0}^{\infty}\frac{H_{2n+1}\left(x\right)}{(2n+1)!}t^{2n+1}}}}$
${\displaystyle{\displaystyle{\displaystyle(1+t^{2})^{-\gamma}\,\HyperpFq{1}{1}@@{\gamma}{\frac{1}{2}}{\frac{x^{2}t^{2}}{1+t^{2}}}=\sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}}}{(2n)!}H_{2n}\left(x\right)t^{2n}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{xt}{\sqrt{1+t^{2}}}\ \HyperpFq{1}{1}@@{\gamma+\frac{1}{2}}{\frac{3}{2}}{\frac{x^{2}t^{2}}{1+t^{2}}}=\sum_{n=0}^{\infty}\frac{{\left(\gamma+\frac{1}{2}\right)_{n}}}{(2n+1)!}H_{2n+1}\left(x\right)t^{2n+1}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{1+2xt+4t^{2}}{(1+4t^{2})^{\frac{3}{2}}}\exp\left(\frac{4x^{2}t^{2}}{1+4t^{2}}\right)=\sum_{n=0}^{\infty}\frac{H_{n}\left(x\right)}{\lfloor n/2\rfloor\,!}t^{n}}}}$

Limit relations

Meixner-Pollaczek polynomial to Hermite polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\lambda\rightarrow\infty}\lambda^{-\frac{1}{2}n}\MeixnerPollaczek{\lambda}{n}@{(\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)}{\phi}=\frac{\Hermite{n}@{x}}{n!}}}}$

Jacobi polynomial to Hermite polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}\alpha^{-\frac{1}{2}n}P^{(\alpha,\alpha)}_{n}\left(\alpha^{-\frac{1}{2}}x\right)=\frac{H_{n}\left(x\right)}{2^{n}n!}}}}$

Gegenbauer / Ultraspherical polynomial to Hermite polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}\alpha^{-\frac{1}{2}n}C^{\alpha+\frac{1}{2}}_{n}\left(\alpha^{-\frac{1}{2}}x\right)=\frac{H_{n}\left(x\right)}{n!}}}}$

Krawtchouk polynomial to Hermite polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}\sqrt{\binomial{N}{n}}K_{n}\!\left(pN+x\sqrt{2p(1-p)N};p,N\right)=\frac{\displaystyle(-1)^{n}H_{n}\left(x\right)}{\displaystyle\sqrt{2^{n}n!\left(\frac{p}{1-p}\right)^{n}}}}}}$

Laguerre polynomial to Hermite polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n}L^{\alpha}_{n}\left((2\alpha\right)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^{n}}{n!}H_{n}\left(x\right)}}}$

Charlier polynomial to Hermite polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow\infty}(2a)^{\frac{1}{2}n}C_{n}\!\left((2a)^{\frac{1}{2}}x+a;a\right)=(-1)^{n}H_{n}\left(x\right)}}}$

Remarks

${\displaystyle{\displaystyle{\displaystyle\frac{H_{n}\left(x\right)}{n!}=\sum_{k=0}^{\lfloor n/2\rfloor}\frac{(-1)^{k}(2x)^{n-2k}}{k!\,(n-2k)!}}}}$
${\displaystyle{\displaystyle{\displaystyle H_{2n}\left(x\right)=(-1)^{n}n!\,2^{2n}L^{-\frac{1}{2}}_{n}\left(x^{2}\right)}}}$
${\displaystyle{\displaystyle{\displaystyle H_{2n+1}\left(x\right)=(-1)^{n}n!\,2^{2n+1}xL^{\frac{1}{2}}_{n}\left(x^{2}\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}H_{n}\left(y\right){\mathrm{e}^{-\frac{1}{2}y^{2}}}{\mathrm{e}^{\mathrm{i}xy}}dy={\mathrm{i}^{n}}H_{n}\left(x\right){\mathrm{e}^{-\frac{1}{2}x^{2}}}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}H_{n}\left(y\right){\mathrm{e}^{-y^{2}}}{\mathrm{e}^{\mathrm{i}xy}}dy={\mathrm{i}^{n}}x^{n}{\mathrm{e}^{-\frac{1}{4}x^{2}}}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{{\mathrm{i}^{n}}}{2\sqrt{\pi}}\int_{-\infty}^{\infty}y^{n}{\mathrm{e}^{-\frac{1}{4}y^{2}}}{\mathrm{e}^{-\mathrm{i}xy}}dy=H_{n}\left(x\right){\mathrm{e}^{-x^{2}}}}}}$

<< Charlier

Hermite

Orthogonal Polynomials >>