Meixner-Pollaczek

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Meixner-Pollaczek

Hypergeometric representation

P n ( λ ) ⁡ ( x ; ϕ ) = ( 2 ⁢ λ ) n n ! ⁢ e i ⁢ n ⁢ ϕ ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ - n , λ + i ⁢ x ⁢ 2 ⁢ λ ⁢ 1 - e - 2 ⁢ i ⁢ ϕ Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ Pochhammer-symbol 2 𝜆 𝑛 𝑛 imaginary-unit 𝑛 italic-ϕ \HyperpFq 21 @ @ 𝑛 𝜆 imaginary-unit 𝑥 2 𝜆 1 2 imaginary-unit italic-ϕ {\displaystyle{\displaystyle{\displaystyle P^{(\lambda)}_{n}\!\left(x;\phi% \right)=\frac{{\left(2\lambda\right)_{n}}}{n!}{\mathrm{e}^{\mathrm{i}n\phi}}\ % \HyperpFq{2}{1}@@{-n,\lambda+\mathrm{i}x}{2\lambda}{1-{\mathrm{e}^{-2\mathrm{i% }\phi}}}}}} {\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)

1 2 ⁢ π ⁢ ∫ - ∞ ∞ e ( 2 ⁢ ϕ - π ) ⁢ x ⁢ | Γ ⁡ ( λ + i ⁢ x ) | 2 ⁢ P m ( λ ) ⁡ ( x ; ϕ ) ⁢ P n ( λ ) ⁡ ( x ; ϕ ) ⁢ 𝑑 x = Γ ⁡ ( n + 2 ⁢ λ ) ( 2 ⁢ sin ⁡ ϕ ) 2 ⁢ λ ⁢ n ! ⁢ δ m , n 1 2 superscript subscript 2 italic-ϕ 𝑥 superscript Euler-Gamma 𝜆 imaginary-unit 𝑥 2 Meixner-Pollaczek-polynomial-P 𝜆 𝑚 𝑥 italic-ϕ Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ differential-d 𝑥 Euler-Gamma 𝑛 2 𝜆 superscript 2 italic-ϕ 2 𝜆 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty% }{\mathrm{e}^{(2\phi-\pi)x}}\left|\Gamma\left(\lambda+\mathrm{i}x\right)\right% |^{2}P^{(\lambda)}_{m}\!\left(x;\phi\right)P^{(\lambda)}_{n}\!\left(x;\phi% \right)\,dx{}=\frac{\Gamma\left(n+2\lambda\right)}{(2\sin\phi)^{2\lambda}n!}\,% \delta_{m,n}}}} {\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): λ > 0   0 < ϕ < π formulae-sequence 𝜆 0 0 italic-ϕ {\displaystyle{\displaystyle{\displaystyle\lambda>0\quad 0<\phi<\pi}}}


Recurrence relation

( n + 1 ) ⁢ P n + 1 ( λ ) ⁡ ( x ; ϕ ) - 2 ⁢ [ x ⁢ sin ⁡ ϕ + ( n + λ ) ⁢ cos ⁡ ϕ ] ⁢ P n ( λ ) ⁡ ( x ; ϕ ) + ( n + 2 ⁢ λ - 1 ) ⁢ P n - 1 ( λ ) ⁡ ( x ; ϕ ) = 0 𝑛 1 Meixner-Pollaczek-polynomial-P 𝜆 𝑛 1 𝑥 italic-ϕ 2 delimited-[] 𝑥 italic-ϕ 𝑛 𝜆 italic-ϕ Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ 𝑛 2 𝜆 1 Meixner-Pollaczek-polynomial-P 𝜆 𝑛 1 𝑥 italic-ϕ 0 {\displaystyle{\displaystyle{\displaystyle(n+1)P^{(\lambda)}_{n+1}\!\left(x;% \phi\right)-2\left[x\sin\phi+(n+\lambda)\cos\phi\right]P^{(\lambda)}_{n}\!% \left(x;\phi\right){}+(n+2\lambda-1)P^{(\lambda)}_{n-1}\!\left(x;\phi\right)=0% }}} {\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

x ⁢ P ^ n ( λ ) ⁡ ( x ) = P ^ n + 1 ( λ ) ⁡ ( x ) - ( n + λ tan ⁡ ϕ ) ⁢ P ^ n ( λ ) ⁡ ( x ) + n ⁢ ( n + 2 ⁢ λ - 1 ) 4 ⁢ sin 2 ⁡ ϕ ⁢ P ^ n - 1 ( λ ) ⁡ ( x ) 𝑥 Meixner-Pollaczek-polynomial-monic-p 𝜆 𝑛 𝑥 italic-ϕ Meixner-Pollaczek-polynomial-monic-p 𝜆 𝑛 1 𝑥 italic-ϕ 𝑛 𝜆 italic-ϕ Meixner-Pollaczek-polynomial-monic-p 𝜆 𝑛 𝑥 italic-ϕ 𝑛 𝑛 2 𝜆 1 4 2 italic-ϕ Meixner-Pollaczek-polynomial-monic-p 𝜆 𝑛 1 𝑥 italic-ϕ {\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}^{(\lambda)}_{n}\!% \left(x\right)={\widehat{P}}^{(\lambda)}_{n+1}\!\left(x\right)-\left(\frac{n+% \lambda}{\tan\phi}\right){\widehat{P}}^{(\lambda)}_{n}\!\left(x\right)+\frac{n% (n+2\lambda-1)}{4{\sin^{2}}\phi}{\widehat{P}}^{(\lambda)}_{n-1}\!\left(x\right% )}}} {\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} }
P n ( λ ) ⁡ ( x ; ϕ ) = ( 2 ⁢ sin ⁡ ϕ ) n n ! ⁢ P ^ n ( λ ) ⁡ ( x ) Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ superscript 2 italic-ϕ 𝑛 𝑛 Meixner-Pollaczek-polynomial-monic-p 𝜆 𝑛 𝑥 italic-ϕ {\displaystyle{\displaystyle{\displaystyle P^{(\lambda)}_{n}\!\left(x;\phi% \right)=\frac{(2\sin\phi)^{n}}{n!}{\widehat{P}}^{(\lambda)}_{n}\!\left(x\right% )}}} {\displaystyle \MeixnerPollaczek{\lambda}{n}@{x}{\phi}=\frac{(2\sin@@{\phi})^n}{n!}\monicMeixnerPollaczek{\lambda}{n}@@{x}{\phi} }

Difference equation

e i ⁢ ϕ ⁢ ( λ - i ⁢ x ) ⁢ y ⁢ ( x + i ) + 2 ⁢ i ⁢ [ x ⁢ cos ⁡ ϕ - ( n + λ ) ⁢ sin ⁡ ϕ ] ⁢ y ⁢ ( x ) - e - i ⁢ ϕ ⁢ ( λ + i ⁢ x ) ⁢ y ⁢ ( x - i ) = 0 imaginary-unit italic-ϕ 𝜆 imaginary-unit 𝑥 𝑦 𝑥 imaginary-unit 2 imaginary-unit delimited-[] 𝑥 italic-ϕ 𝑛 𝜆 italic-ϕ 𝑦 𝑥 imaginary-unit italic-ϕ 𝜆 imaginary-unit 𝑥 𝑦 𝑥 imaginary-unit 0 {\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{\mathrm{i}\phi}}(% \lambda-\mathrm{i}x)y(x+\mathrm{i})+2\mathrm{i}\left[x\cos\phi-(n+\lambda)\sin% \phi\right]y(x){}-{\mathrm{e}^{-\mathrm{i}\phi}}(\lambda+\mathrm{i}x)y(x-% \mathrm{i})=0}}} {\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): y ⁢ ( x ) = P n ( λ ) ⁡ ( x ; ϕ ) 𝑦 𝑥 Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ {\displaystyle{\displaystyle{\displaystyle y(x)=P^{(\lambda)}_{n}\!\left(x;% \phi\right)}}}


Forward shift operator

P n ( λ ) ⁡ ( x + 1 2 ⁢ i ; ϕ ) - P n ( λ ) ⁡ ( x - 1 2 ⁢ i ; ϕ ) = ( e i ⁢ ϕ - e - i ⁢ ϕ ) ⁢ P n - 1 ( λ + 1 2 ) ⁡ ( x ; ϕ ) Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 1 2 imaginary-unit italic-ϕ Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 1 2 imaginary-unit italic-ϕ imaginary-unit italic-ϕ imaginary-unit italic-ϕ Meixner-Pollaczek-polynomial-P 𝜆 1 2 𝑛 1 𝑥 italic-ϕ {\displaystyle{\displaystyle{\displaystyle P^{(\lambda)}_{n}\!\left(x+% \textstyle\frac{1}{2}\mathrm{i};\phi\right)-P^{(\lambda)}_{n}\!\left(x-% \textstyle\frac{1}{2}\mathrm{i};\phi\right)=({\mathrm{e}^{\mathrm{i}\phi}}-{% \mathrm{e}^{-\mathrm{i}\phi}})P^{(\lambda+\frac{1}{2})}_{n-1}\!\left(x;\phi% \right)}}} {\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} }
δ ⁢ P n ( λ ) ⁡ ( x ; ϕ ) δ ⁢ x = 2 ⁢ sin ⁡ ϕ ⁢ P n - 1 ( λ + 1 2 ) ⁡ ( x ; ϕ ) 𝛿 Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ 𝛿 𝑥 2 italic-ϕ Meixner-Pollaczek-polynomial-P 𝜆 1 2 𝑛 1 𝑥 italic-ϕ {\displaystyle{\displaystyle{\displaystyle\frac{\delta P^{(\lambda)}_{n}\!% \left(x;\phi\right)}{\delta x}=2\sin\phi\,P^{(\lambda+\frac{1}{2})}_{n-1}\!% \left(x;\phi\right)}}} {\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

e i ⁢ ϕ ⁢ ( λ - 1 2 - i ⁢ x ) ⁢ P n ( λ ) ⁡ ( x + 1 2 ⁢ i ; ϕ ) + e - i ⁢ ϕ ⁢ ( λ - 1 2 + i ⁢ x ) ⁢ P n ( λ ) ⁡ ( x - 1 2 ⁢ i ; ϕ ) = ( n + 1 ) ⁢ P n + 1 ( λ - 1 2 ) ⁡ ( x ; ϕ ) imaginary-unit italic-ϕ 𝜆 1 2 imaginary-unit 𝑥 Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 1 2 imaginary-unit italic-ϕ imaginary-unit italic-ϕ 𝜆 1 2 imaginary-unit 𝑥 Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 1 2 imaginary-unit italic-ϕ 𝑛 1 Meixner-Pollaczek-polynomial-P 𝜆 1 2 𝑛 1 𝑥 italic-ϕ {\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{\mathrm{i}\phi}}(% \lambda-\textstyle\frac{1}{2}-\mathrm{i}x)P^{(\lambda)}_{n}\!\left(x+% \textstyle\frac{1}{2}\mathrm{i};\phi\right)+{\mathrm{e}^{-\mathrm{i}\phi}}(% \lambda-\textstyle\frac{1}{2}+\mathrm{i}x)P^{(\lambda)}_{n}\!\left(x-% \textstyle\frac{1}{2}\mathrm{i};\phi\right){}=(n+1)P^{(\lambda-\frac{1}{2})}_{% n+1}\!\left(x;\phi\right)}}} {\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} }
δ ⁢ [ ω ⁢ ( x ; λ , ϕ ) ⁢ P n ( λ ) ⁡ ( x ; ϕ ) ] δ ⁢ x = - ( n + 1 ) ⁢ ω ⁢ ( x ; λ - 1 2 , ϕ ) ⁢ P n + 1 ( λ - 1 2 ) ⁡ ( x ; ϕ ) 𝛿 delimited-[] 𝜔 𝑥 𝜆 italic-ϕ Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ 𝛿 𝑥 𝑛 1 𝜔 𝑥 𝜆 1 2 italic-ϕ Meixner-Pollaczek-polynomial-P 𝜆 1 2 𝑛 1 𝑥 italic-ϕ {\displaystyle{\displaystyle{\displaystyle\frac{\delta\left[\omega(x;\lambda,% \phi)P^{(\lambda)}_{n}\!\left(x;\phi\right)\right]}{\delta x}=-(n+1)\omega(x;% \lambda-\textstyle\frac{1}{2},\phi)P^{(\lambda-\frac{1}{2})}_{n+1}\!\left(x;% \phi\right)}}} {\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): ω ⁢ ( x ; λ , ϕ ) = Γ ⁡ ( λ + i ⁢ x ) ⁢ Γ ⁡ ( λ - i ⁢ x ) ⁢ e ( 2 ⁢ ϕ - π ) ⁢ x 𝜔 𝑥 𝜆 italic-ϕ Euler-Gamma 𝜆 imaginary-unit 𝑥 Euler-Gamma 𝜆 imaginary-unit 𝑥 2 italic-ϕ 𝑥 {\displaystyle{\displaystyle{\displaystyle\omega(x;\lambda,\phi)=\Gamma\left(% \lambda+\mathrm{i}x\right)\Gamma\left(\lambda-\mathrm{i}x\right){\mathrm{e}^{(% 2\phi-\pi)x}}}}}


Rodrigues-type formula

ω ⁢ ( x ; λ , ϕ ) ⁢ P n ( λ ) ⁡ ( x ; ϕ ) = ( - 1 ) n n ! ⁢ ( δ δ ⁢ x ) n ⁢ [ ω ⁢ ( x ; λ + 1 2 ⁢ n , ϕ ) ] 𝜔 𝑥 𝜆 italic-ϕ Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ superscript 1 𝑛 𝑛 superscript 𝛿 𝛿 𝑥 𝑛 delimited-[] 𝜔 𝑥 𝜆 1 2 𝑛 italic-ϕ {\displaystyle{\displaystyle{\displaystyle\omega(x;\lambda,\phi)P^{(\lambda)}_% {n}\!\left(x;\phi\right)=\frac{(-1)^{n}}{n!}\left(\frac{\delta}{\delta x}% \right)^{n}\left[\omega(x;\lambda+\textstyle\frac{1}{2}n,\phi)\right]}}} {\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): ω ⁢ ( x ; λ , ϕ ) = Γ ⁡ ( λ + i ⁢ x ) ⁢ Γ ⁡ ( λ - i ⁢ x ) ⁢ e ( 2 ⁢ ϕ - π ) ⁢ x 𝜔 𝑥 𝜆 italic-ϕ Euler-Gamma 𝜆 imaginary-unit 𝑥 Euler-Gamma 𝜆 imaginary-unit 𝑥 2 italic-ϕ 𝑥 {\displaystyle{\displaystyle{\displaystyle\omega(x;\lambda,\phi)=\Gamma\left(% \lambda+\mathrm{i}x\right)\Gamma\left(\lambda-\mathrm{i}x\right){\mathrm{e}^{(% 2\phi-\pi)x}}}}}


Generating functions

( 1 - e i ⁢ ϕ ⁢ t ) - λ + i ⁢ x ⁢ ( 1 - e - i ⁢ ϕ ⁢ t ) - λ - i ⁢ x = ∑ n = 0 ∞ P n ( λ ) ⁡ ( x ; ϕ ) ⁢ t n superscript 1 imaginary-unit italic-ϕ 𝑡 𝜆 imaginary-unit 𝑥 superscript 1 imaginary-unit italic-ϕ 𝑡 𝜆 imaginary-unit 𝑥 superscript subscript 𝑛 0 Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle(1-{\mathrm{e}^{\mathrm{i}\phi}}t)^{% -\lambda+\mathrm{i}x}(1-{\mathrm{e}^{-\mathrm{i}\phi}}t)^{-\lambda-\mathrm{i}x% }=\sum_{n=0}^{\infty}P^{(\lambda)}_{n}\!\left(x;\phi\right)t^{n}}}} {\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 }
e t ⁢ \HyperpFq ⁢ 11 ⁢ @ ⁢ @ ⁢ λ + i ⁢ x ⁢ 2 ⁢ λ ⁢ ( e - 2 ⁢ i ⁢ ϕ - 1 ) ⁢ t = ∑ n = 0 ∞ P n ( λ ) ⁡ ( x ; ϕ ) ( 2 ⁢ λ ) n ⁢ e i ⁢ n ⁢ ϕ ⁢ t n 𝑡 \HyperpFq 11 @ @ 𝜆 imaginary-unit 𝑥 2 𝜆 2 imaginary-unit italic-ϕ 1 𝑡 superscript subscript 𝑛 0 Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ Pochhammer-symbol 2 𝜆 𝑛 imaginary-unit 𝑛 italic-ϕ superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{t}}\,\HyperpFq{1}{1}@@{% \lambda+\mathrm{i}x}{2\lambda}{({\mathrm{e}^{-2\mathrm{i}\phi}}-1)t}=\sum_{n=0% }^{\infty}\frac{P^{(\lambda)}_{n}\!\left(x;\phi\right)}{{\left(2\lambda\right)% _{n}}{\mathrm{e}^{\mathrm{i}n\phi}}}t^{n}}}} {\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 }
( 1 - t ) - γ ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ γ , λ + i ⁢ x ⁢ 2 ⁢ λ ⁢ ( 1 - e - 2 ⁢ i ⁢ ϕ ) ⁢ t t - 1 = ∑ n = 0 ∞ ( γ ) n ( 2 ⁢ λ ) n ⁢ P n ( λ ) ⁡ ( x ; ϕ ) e i ⁢ n ⁢ ϕ ⁢ t n superscript 1 𝑡 𝛾 \HyperpFq 21 @ @ 𝛾 𝜆 imaginary-unit 𝑥 2 𝜆 1 2 imaginary-unit italic-ϕ 𝑡 𝑡 1 superscript subscript 𝑛 0 Pochhammer-symbol 𝛾 𝑛 Pochhammer-symbol 2 𝜆 𝑛 Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ imaginary-unit 𝑛 italic-ϕ superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle(1-t)^{-\gamma}\,\HyperpFq{2}{1}@@{% \gamma,\lambda+\mathrm{i}x}{2\lambda}{\frac{(1-{\mathrm{e}^{-2\mathrm{i}\phi}}% )t}{t-1}}{}=\sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}}}{{\left(2% \lambda\right)_{n}}}\frac{P^{(\lambda)}_{n}\!\left(x;\phi\right)}{{\mathrm{e}^% {\mathrm{i}n\phi}}}t^{n}}}} {\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): γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


Limit relations

Continuous dual Hahn polynomial to Meixner-Pollaczek polynomial

lim t → ∞ ⁡ S n ⁡ ( ( x - t ) 2 ; λ + i ⁢ t , λ - i ⁢ t , t ⁢ cot ⁡ ϕ ) t n ⁢ n ! = P n ( λ ) ⁡ ( x ; ϕ ) ( sin ⁡ ϕ ) n subscript → 𝑡 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 𝑡 2 𝜆 imaginary-unit 𝑡 𝜆 imaginary-unit 𝑡 𝑡 italic-ϕ superscript 𝑡 𝑛 𝑛 Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ superscript italic-ϕ 𝑛 {\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}}}}} {\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

lim t → ∞ ⁡ p n ⁡ ( x + t ; λ - i ⁢ t , t ⁢ tan ⁡ ϕ , λ + i ⁢ t , t ⁢ tan ⁡ ϕ ) t n ⁢ n ! = P n ( λ ) ⁡ ( x ; ϕ ) ( cos ⁡ ϕ ) n subscript → 𝑡 continuous-Hahn-polynomial 𝑛 𝑥 𝑡 𝜆 imaginary-unit 𝑡 𝑡 italic-ϕ 𝜆 imaginary-unit 𝑡 𝑡 italic-ϕ superscript 𝑡 𝑛 𝑛 Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ superscript italic-ϕ 𝑛 {\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}\frac{p_{n}% \!\left(x+t;\lambda-\mathrm{i}t,t\tan\phi,\lambda+\mathrm{i}t,t\tan\phi\right)% }{t^{n}n!}=\frac{P^{(\lambda)}_{n}\!\left(x;\phi\right)}{(\cos\phi)^{n}}}}} {\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

lim ϕ → 0 ⁡ P n ( 1 2 ⁢ α + 1 2 ) ⁡ ( - 1 2 ⁢ ϕ - 1 ⁢ x ; ϕ ) = L n α ⁡ ( x ) subscript → italic-ϕ 0 Meixner-Pollaczek-polynomial-P 1 2 𝛼 1 2 𝑛 1 2 superscript italic-ϕ 1 𝑥 italic-ϕ generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\lim_{\phi\rightarrow 0}P^{(\frac{1}% {2}\alpha+\frac{1}{2})}_{n}\!\left(-\textstyle\frac{1}{2}\phi^{-1}x;\phi\right% )=L^{\alpha}_{n}\left(x\right)}}} {\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

lim λ → ∞ ⁡ λ - 1 2 ⁢ n ⁢ P n ( λ ) ⁡ ( ( sin ⁡ ϕ ) - 1 ⁢ ( x ⁢ λ - λ ⁢ cos ⁡ ϕ ) ; ϕ ) = H n ⁡ ( x ) n ! subscript → 𝜆 superscript 𝜆 1 2 𝑛 Meixner-Pollaczek-polynomial-P 𝜆 𝑛 superscript italic-ϕ 1 𝑥 𝜆 𝜆 italic-ϕ italic-ϕ Hermite-polynomial-H 𝑛 𝑥 𝑛 {\displaystyle{\displaystyle{\displaystyle\lim_{\lambda\rightarrow\infty}% \lambda^{-\frac{1}{2}n}P^{(\lambda)}_{n}\!\left((\sin\phi)^{-1}(x\sqrt{\lambda% }-\lambda\cos\phi);\phi\right)=\frac{H_{n}\left(x\right)}{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!} }

Remark

( 2 ⁢ λ ) n ( 2 ⁢ λ ) k = ( 2 ⁢ λ + k ) n - k Pochhammer-symbol 2 𝜆 𝑛 Pochhammer-symbol 2 𝜆 𝑘 Pochhammer-symbol 2 𝜆 𝑘 𝑛 𝑘 {\displaystyle{\displaystyle{\displaystyle\frac{{\left(2\lambda\right)_{n}}}{{% \left(2\lambda\right)_{k}}}={\left(2\lambda+k\right)_{n-k}}}}} {\displaystyle \frac{\pochhammer{2\lambda}{n}}{\pochhammer{2\lambda}{k}}=\pochhammer{2\lambda+k}{n-k} }

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