Meixner-Pollaczek

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Meixner-Pollaczek

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Hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle\MeixnerPollaczek{\lambda}{n}@{x}{\phi}=\frac{\pochhammer{2\lambda}{n}}{n!}\expe^{\iunit n\phi}\ {{}_{2}F_{1}}\!\left({-n,\lambda+\iunit x\atop 2\lambda};1-\expe^{-2\iunit\phi}\right)}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\cpi}\int_{-\infty}^{\infty}\expe^{(2\phi-\cpi)x}\left|\Gamma\left(\lambda+\iunit x\right)\right|^{2}\MeixnerPollaczek{\lambda}{m}@{x}{\phi}\MeixnerPollaczek{\lambda}{n}@{x}{\phi}\,dx{}=\frac{\Gamma\left(n+2\lambda\right)}{(2\sin\phi)^{2\lambda}n!}\,\Kronecker{m}{n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\lambda>0\quad 0<\phi<\cpi}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle(n+1)\MeixnerPollaczek{\lambda}{n+1}@{x}{\phi}-2\left[x\sin\phi+(n+\lambda)\cos\phi\right]\MeixnerPollaczek{\lambda}{n}@{x}{\phi}{}+(n+2\lambda-1)\MeixnerPollaczek{\lambda}{n-1}@{x}{\phi}=0}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x\monicMeixnerPollaczek{\lambda}{n}@@{x}{\phi}=\monicMeixnerPollaczek{\lambda}{n+1}@@{x}{\phi}-\left(\frac{n+\lambda}{\tan\phi}\right)\monicMeixnerPollaczek{\lambda}{n}@@{x}{\phi}+\frac{n(n+2\lambda-1)}{4{\sin^{2}}\phi}\monicMeixnerPollaczek{\lambda}{n-1}@@{x}{\phi}}}}$
${\displaystyle{\displaystyle{\displaystyle\MeixnerPollaczek{\lambda}{n}@{x}{\phi}=\frac{(2\sin\phi)^{n}}{n!}\monicMeixnerPollaczek{\lambda}{n}@@{x}{\phi}}}}$

Difference equation

${\displaystyle{\displaystyle{\displaystyle\expe^{\iunit\phi}(\lambda-\iunit x)y(x+\iunit)+2\iunit\left[x\cos\phi-(n+\lambda)\sin\phi\right]y(x){}-\expe^{-\iunit\phi}(\lambda+\iunit x)y(x-\iunit)=0}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle y(x)=\MeixnerPollaczek{\lambda}{n}@{x}{\phi}}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle\MeixnerPollaczek{\lambda}{n}@{x+\textstyle\frac{1}{2}\iunit}{\phi}-\MeixnerPollaczek{\lambda}{n}@{x-\textstyle\frac{1}{2}\iunit}{\phi}=(\expe^{\iunit\phi}-\expe^{-\iunit\phi})\MeixnerPollaczek{\lambda+\frac{1}{2}}{n-1}@{x}{\phi}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{\delta\MeixnerPollaczek{\lambda}{n}@{x}{\phi}}{\delta x}=2\sin\phi\,\MeixnerPollaczek{\lambda+\frac{1}{2}}{n-1}@{x}{\phi}}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle\expe^{\iunit\phi}(\lambda-\textstyle\frac{1}{2}-\iunit x)\MeixnerPollaczek{\lambda}{n}@{x+\textstyle\frac{1}{2}\iunit}{\phi}+\expe^{-\iunit\phi}(\lambda-\textstyle\frac{1}{2}+\iunit x)\MeixnerPollaczek{\lambda}{n}@{x-\textstyle\frac{1}{2}\iunit}{\phi}{}=(n+1)\MeixnerPollaczek{\lambda-\frac{1}{2}}{n+1}@{x}{\phi}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{\delta\left[\omega(x;\lambda,\phi)\MeixnerPollaczek{\lambda}{n}@{x}{\phi}\right]}{\delta x}=-(n+1)\omega(x;\lambda-\textstyle\frac{1}{2},\phi)\MeixnerPollaczek{\lambda-\frac{1}{2}}{n+1}@{x}{\phi}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\omega(x;\lambda,\phi)=\Gamma\left(\lambda+\iunit x\right)\Gamma\left(\lambda-\iunit x\right)\expe^{(2\phi-\cpi)x}}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle\omega(x;\lambda,\phi)\MeixnerPollaczek{\lambda}{n}@{x}{\phi}=\frac{(-1)^{n}}{n!}\left(\frac{\delta}{\delta x}\right)^{n}\left[\omega(x;\lambda+\textstyle\frac{1}{2}n,\phi)\right]}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\omega(x;\lambda,\phi)=\Gamma\left(\lambda+\iunit x\right)\Gamma\left(\lambda-\iunit x\right)\expe^{(2\phi-\cpi)x}}}}$

Generating functions

${\displaystyle{\displaystyle{\displaystyle(1-\expe^{\iunit\phi}t)^{-\lambda+\iunit x}(1-\expe^{-\iunit\phi}t)^{-\lambda-\iunit x}=\sum_{n=0}^{\infty}\MeixnerPollaczek{\lambda}{n}@{x}{\phi}t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\expe^{t}\,{{}_{1}F_{1}}\!\left({\lambda+\iunit x\atop 2\lambda};(\expe^{-2\iunit\phi}-1)t\right)=\sum_{n=0}^{\infty}\frac{\MeixnerPollaczek{\lambda}{n}@{x}{\phi}}{\pochhammer{2\lambda}{n}\expe^{\iunit n\phi}}t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle(1-t)^{-\gamma}\,{{}_{2}F_{1}}\!\left({\gamma,\lambda+\iunit x\atop 2\lambda};\frac{(1-\expe^{-2\iunit\phi})t}{t-1}\right){}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}}{\pochhammer{2\lambda}{n}}\frac{\MeixnerPollaczek{\lambda}{n}@{x}{\phi}}{\expe^{\iunit n\phi}}t^{n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\gamma}}}$ arbitrary

Limit relations

Continuous dual Hahn polynomial to Meixner-Pollaczek polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}\frac{S_{n}\!\left((x-t)^{2};\lambda+\mathrm{i}t,\lambda-\mathrm{i}t,t\cot\phi\right)}{t^{n}n!}=\frac{P^{(\lambda)}_{n}\!\left(x;\phi\right)}{(\sin\phi)^{n}}}}}$

Continuous Hahn polynomial to Meixner-Pollaczek polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}\frac{\ctsHahn{n}@{x+t}{\lambda-\iunit t}{t\tan\phi}{\lambda+\iunit t}{t\tan\phi}}{t^{n}n!}=\frac{\MeixnerPollaczek{\lambda}{n}@{x}{\phi}}{(\cos\phi)^{n}}}}}$

Meixner-Pollaczek polynomial to Laguerre polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\phi\rightarrow 0}\MeixnerPollaczek{\frac{1}{2}\alpha+\frac{1}{2}}{n}@{-\textstyle\frac{1}{2}\phi^{-1}x}{\phi}=\Laguerre[\alpha]{n}@{x}}}}$

Meixner-Pollaczek polynomial to Hermite polynomial

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

Remark

${\displaystyle{\displaystyle{\displaystyle\frac{\pochhammer{2\lambda}{n}}{\pochhammer{2\lambda}{k}}=\pochhammer{2\lambda+k}{n-k}}}}$

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Meixner-Pollaczek

Jacobi >>