Meixner-Pollaczek

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Meixner-Pollaczek

Jacobi >>

Hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \MeixnerPollaczek{\lambda}{n}@{x}{\phi}=\frac{\pochhammer{2\lambda}{n}}{n!}\expe^{\iunit n\phi}\ \HyperpFq{2}{1}@@{-n,\lambda+\iunit x}{2\lambda}{1-\expe^{-2\iunit\phi}} }}

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{2\cpi}\int_{-\infty}^{\infty} \expe^{(2\phi-\cpi)x}\left|\EulerGamma@{\lambda+\iunit x}\right|^2 \MeixnerPollaczek{\lambda}{m}@{x}{\phi}\MeixnerPollaczek{\lambda}{n}@{x}{\phi}\,dx {}=\frac{\EulerGamma@{n+2\lambda}}{(2\sin@@{\phi})^{2\lambda}n!}\,\Kronecker{m}{n} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda > 0\quad 0 < \phi < \cpi}}

Recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (n+1)\MeixnerPollaczek{\lambda}{n+1}@{x}{\phi}-2\left[x\sin@@{\phi}+(n+\lambda)\cos@@{\phi}\right] \MeixnerPollaczek{\lambda}{n}@{x}{\phi} {}+(n+2\lambda-1)\MeixnerPollaczek{\lambda}{n-1}@{x}{\phi}=0 }}

Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicMeixnerPollaczek{\lambda}{n}@@{x}{\phi}=\monicMeixnerPollaczek{\lambda}{n+1}@@{x}{\phi}-\left(\frac{n+\lambda}{\tan@@{\phi}}\right)\monicMeixnerPollaczek{\lambda}{n}@@{x}{\phi}+ \frac{n(n+2\lambda-1)}{4\sin^2@@{\phi}}\monicMeixnerPollaczek{\lambda}{n-1}@@{x}{\phi} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \MeixnerPollaczek{\lambda}{n}@{x}{\phi}=\frac{(2\sin@@{\phi})^n}{n!}\monicMeixnerPollaczek{\lambda}{n}@@{x}{\phi} }}

Difference equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \expe^{\iunit\phi}(\lambda-\iunit x)y(x+\iunit)+2\iunit\left[x\cos@@{\phi}-(n+\lambda)\sin@@{\phi}\right]y(x) {}-\expe^{-\iunit\phi}(\lambda+\iunit x)y(x-\iunit)=0 }}

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

Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \MeixnerPollaczek{\lambda}{n}@{x+\textstyle\frac{1}{2}\iunit}{\phi}- \MeixnerPollaczek{\lambda}{n}@{x-\textstyle\frac{1}{2}\iunit}{\phi}=(\expe^{\iunit\phi}-\expe^{-\iunit\phi}) \MeixnerPollaczek{\lambda+\frac{1}{2}}{n-1}@{x}{\phi} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\delta \MeixnerPollaczek{\lambda}{n}@{x}{\phi}}{\delta x} =2\sin@@{\phi}\,\MeixnerPollaczek{\lambda+\frac{1}{2}}{n-1}@{x}{\phi} }}

Backward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \expe^{\iunit\phi}(\lambda-\textstyle\frac{1}{2}-\iunit x) \MeixnerPollaczek{\lambda}{n}@{x+\textstyle\frac{1}{2}\iunit}{\phi}+ \expe^{-\iunit\phi}(\lambda-\textstyle\frac{1}{2}+\iunit x) \MeixnerPollaczek{\lambda}{n}@{x-\textstyle\frac{1}{2}\iunit}{\phi} {}=(n+1)\MeixnerPollaczek{\lambda-\frac{1}{2}}{n+1}@{x}{\phi} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\delta\left[\omega(x;\lambda,\phi)\MeixnerPollaczek{\lambda}{n}@{x}{\phi}\right]}{\delta x}= -(n+1)\omega(x;\lambda-\textstyle\frac{1}{2},\phi) \MeixnerPollaczek{\lambda-\frac{1}{2}}{n+1}@{x}{\phi} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \omega(x;\lambda,\phi)=\EulerGamma@{\lambda+\iunit x}\EulerGamma@{\lambda-\iunit x}\expe^{(2\phi-\cpi)x}}}

Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \omega(x;\lambda,\phi)\MeixnerPollaczek{\lambda}{n}@{x}{\phi}=\frac{(-1)^n}{n!} \left(\frac{\delta}{\delta x}\right)^n\left[\omega(x;\lambda+\textstyle\frac{1}{2}n,\phi)\right] }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \omega(x;\lambda,\phi)=\EulerGamma@{\lambda+\iunit x}\EulerGamma@{\lambda-\iunit x}\expe^{(2\phi-\cpi)x}}}

Generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-\expe^{\iunit\phi}t)^{-\lambda+\iunit x}(1-\expe^{-\iunit\phi}t)^{-\lambda-\iunit x}= \sum_{n=0}^{\infty}\MeixnerPollaczek{\lambda}{n}@{x}{\phi}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \expe^t\,\HyperpFq{1}{1}@@{\lambda+\iunit x}{2\lambda}{(\expe^{-2\iunit\phi}-1)t}= \sum_{n=0}^{\infty}\frac{\MeixnerPollaczek{\lambda}{n}@{x}{\phi}}{\pochhammer{2\lambda}{n}\expe^{\iunit n\phi}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-t)^{-\gamma}\,\HyperpFq{2}{1}@@{\gamma,\lambda+\iunit x}{2\lambda}{\frac{(1-\expe^{-2\iunit\phi})t}{t-1}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}}{\pochhammer{2\lambda}{n}}\frac{\MeixnerPollaczek{\lambda}{n}@{x}{\phi}}{\expe^{\iunit n\phi}}t^n }}

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

Limit relations

Continuous dual Hahn polynomial to Meixner-Pollaczek polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{t\rightarrow\infty}\frac{\ctsdualHahn{n}@{(x-t)^2}{\lambda+\iunit t}{\lambda-\iunit t}{t\cot@@{\phi}}}{t^nn!} =\frac{\MeixnerPollaczek{\lambda}{n}@{x}{\phi}}{(\sin@@{\phi})^n} }}

Continuous Hahn polynomial to Meixner-Pollaczek polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{t\rightarrow\infty}\frac{\ctsHahn{n}@{x+t}{\lambda-\iunit t}{t\tan@@{\phi}}{\lambda+\iunit t}{t\tan@@{\phi}}}{t^nn!} =\frac{\MeixnerPollaczek{\lambda}{n}@{x}{\phi}}{(\cos@@{\phi})^n} }}

Meixner-Pollaczek polynomial to Laguerre polynomial

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

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!} }}

Remark

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\pochhammer{2\lambda}{n}}{\pochhammer{2\lambda}{k}}=\pochhammer{2\lambda+k}{n-k} }}

<< Dual Hahn

Meixner-Pollaczek

Jacobi >>