Hermite

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Hermite

Hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hermite{n}@{x}=(2x)^n\,\HyperpFq{2}{0}@@{-n/2,-(n-1)/2}{-}{-\frac{1}{x^2}} }}

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hermite{n+1}@{x}-2x\Hermite{n}@{x}+2n\Hermite{n-1}@{x}=0 }}

Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicHermite{n}@@{x}{x}=\monicHermite{n+1}@@{x}{x}+\frac{n}{2}\monicHermite{n-1}@@{x}{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hermite{n}@{x}=2^n\monicHermite{n}@@{x}{x} }}

Differential equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y''(x)-2xy'(x)+2ny(x)=0 }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\Hermite{n}@{x}}}


Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{d}{dx}\Hermite{n}@{x}=2n\Hermite{n-1}@{x} }}

Backward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{d}{dx}\Hermite{n}@{x}-2x\Hermite{n}@{x}=-\Hermite{n+1}@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{d}{dx}\left[\expe^{-x^2}\Hermite{n}@{x}\right]=-\expe^{-x^2}\Hermite{n+1}@{x} }}

Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \expe^{-x^2}\Hermite{n}@{x}=(-1)^n\left(\frac{d}{dx}\right)^n\left[\expe^{-x^2}\right] }}

Generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \exp@{2xt-t^2}=\sum_{n=0}^{\infty}\frac{\Hermite{n}@{x}}{n!}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \expe^t\cos@{2x\sqrt{t}}=\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}\Hermite{2n}@{x}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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 }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \expe^{-t^2}\cosh@{2xt}=\sum_{n=0}^{\infty} \frac{\Hermite{2n}@{x}}{(2n)!}t^{2n} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \expe^{-t^2}\sinh@{2xt}=\sum_{n=0}^{\infty} \frac{\Hermite{2n+1}@{x}}{(2n+1)!}t^{2n+1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\Hermite{n}@{x}}{n!}=\sum_{k=0}^{\lfloor n/2\rfloor} \frac{(-1)^k(2x)^{n-2k}}{k!\,(n-2k)!} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hermite{2n}@{x}=(-1)^nn!\,2^{2n}\Laguerre[-\frac{1}{2}]{n}@{x^2} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hermite{2n+1}@{x}=(-1)^nn!\,2^{2n+1}x\Laguerre[\frac{1}{2}]{n}@{x^2} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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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