q-Meixner-Pollaczek

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q-Meixner-Pollaczek

Basic hypergeometric representation

P n ⁡ ( x ; a | q ) = a - n ⁢ e - i ⁢ n ⁢ ϕ ⁢ ( a 2 ; q ) n ( q ; q ) n ⁢ \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ q - n , a ⁢ e i ⁢ ( θ + 2 ⁢ ϕ ) , a ⁢ e - i ⁢ θ ⁢ a 2 , 0 ⁢ q ⁢ q q-Meixner-Pollaczek-polynomial-P 𝑛 𝑥 𝑎 𝑞 superscript 𝑎 𝑛 imaginary-unit 𝑛 italic-ϕ q-Pochhammer-symbol superscript 𝑎 2 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 𝑎 imaginary-unit 𝜃 2 italic-ϕ 𝑎 imaginary-unit 𝜃 superscript 𝑎 2 0 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a|q\right)=a^{-n}{% \mathrm{e}^{-\mathrm{i}n\phi}}\frac{\left(a^{2};q\right)_{n}}{\left(q;q\right)% _{n}}\ \qHyperrphis{3}{2}@@{q^{-n},a{\mathrm{e}^{\mathrm{i}(\theta+2\phi)}},a{% \mathrm{e}^{-\mathrm{i}\theta}}}{a^{2},0}{q}{q}}}} {\displaystyle \qMeixnerPollaczek{n}@{x}{a}{q}=a^{-n}\expe^{-\iunit n\phi}\frac{\qPochhammer{a^2}{q}{n}}{\qPochhammer{q}{q}{n}}\ \qHyperrphis{3}{2}@@{q^{-n},a\expe^{\iunit(\theta+2\phi)},a\expe^{-\iunit\theta}}{a^2,0}{q}{q} }
P n ⁡ ( x ; a | q ) = ( a ⁢ e - i ⁢ θ ; q ) n ( q ; q ) n ⁢ e i ⁢ n ⁢ ( θ + ϕ ) ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - n , a ⁢ e i ⁢ θ ⁢ a - 1 ⁢ q - n + 1 ⁢ e i ⁢ θ ⁢ q ⁢ q ⁢ a - 1 ⁢ e - i ⁢ ( θ + 2 ⁢ ϕ ) q-Meixner-Pollaczek-polynomial-P 𝑛 𝑥 𝑎 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 imaginary-unit 𝑛 𝜃 italic-ϕ \qHyperrphis 21 @ @ superscript 𝑞 𝑛 𝑎 imaginary-unit 𝜃 superscript 𝑎 1 superscript 𝑞 𝑛 1 imaginary-unit 𝜃 𝑞 𝑞 superscript 𝑎 1 imaginary-unit 𝜃 2 italic-ϕ {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a|q\right)=\frac{% \left(a{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{n}}{\left(q;q\right)_{n}}{% \mathrm{e}^{\mathrm{i}n(\theta+\phi)}}\ \qHyperrphis{2}{1}@@{q^{-n},a{\mathrm{% e}^{\mathrm{i}\theta}}}{a^{-1}q^{-n+1}{\mathrm{e}^{\mathrm{i}\theta}}}{q}{qa^{% -1}{\mathrm{e}^{-\mathrm{i}(\theta+2\phi)}}}}}} {\displaystyle \qMeixnerPollaczek{n}@{x}{a}{q}=\frac{\qPochhammer{a\expe^{-\iunit\theta}}{q}{n}}{\qPochhammer{q}{q}{n}}\expe^{\iunit n(\theta+\phi)}\ \qHyperrphis{2}{1}@@{q^{-n},a\expe^{\iunit\theta}}{a^{-1}q^{-n+1}\expe^{\iunit\theta}}{q}{qa^{-1}\expe^{-\iunit(\theta+2\phi)}} }

Orthogonality relation(s)

1 2 ⁢ π ⁢ ∫ - π π w ⁢ ( cos ⁡ ( θ + ϕ ) ; a | q ) ⁢ P m ⁡ ( cos ⁡ ( θ + ϕ ) ; a | q ) ⁢ P n ⁡ ( cos ⁡ ( θ + ϕ ) ; a | q ) ⁢ 𝑑 θ = δ m , n ( q ; q ) n ⁢ ( q , a 2 ⁢ q n ; q ) ∞ 1 2 superscript subscript 𝑤 𝜃 italic-ϕ conditional 𝑎 𝑞 q-Meixner-Pollaczek-polynomial-P 𝑚 𝜃 italic-ϕ 𝑎 𝑞 q-Meixner-Pollaczek-polynomial-P 𝑛 𝜃 italic-ϕ 𝑎 𝑞 differential-d 𝜃 Kronecker-delta 𝑚 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑞 superscript 𝑎 2 superscript 𝑞 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}w(% \cos\left(\theta+\phi\right);a|q)P_{m}\!\left(\cos\left(\theta+\phi\right);a|q% \right)P_{n}\!\left(\cos\left(\theta+\phi\right);a|q\right)\,d\theta{}=\frac{% \,\delta_{m,n}}{\left(q;q\right)_{n}\left(q,a^{2}q^{n};q\right)_{\infty}}}}} {\displaystyle \frac{1}{2\cpi}\int_{-\cpi}^{\cpi}w(\cos@{\theta+\phi};a|q) \qMeixnerPollaczek{m}@{\cos@{\theta+\phi}}{a}{q}\qMeixnerPollaczek{n}@{\cos@{\theta+\phi}}{a}{q}\,d\theta {}=\frac{\,\Kronecker{m}{n}}{\qPochhammer{q}{q}{n}\qPochhammer{q,a^2q^n}{q}{\infty}} }

Constraint(s): 0 < a < 1 0 𝑎 1 {\displaystyle{\displaystyle{\displaystyle 0<a<1}}}


Substitution(s): w ⁢ ( x ; a | q ) = | ( e 2 ⁢ i ⁢ ( θ + ϕ ) ; q ) ∞ ( a ⁢ e i ⁢ ( θ + 2 ⁢ ϕ ) , a ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ⁢ e i ⁢ ϕ ) ⁢ h ⁢ ( x , a ⁢ e - i ⁢ ϕ ) 𝑤 𝑥 conditional 𝑎 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 italic-ϕ 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 2 italic-ϕ 𝑎 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 imaginary-unit italic-ϕ ℎ 𝑥 𝑎 imaginary-unit italic-ϕ {\displaystyle{\displaystyle{\displaystyle w(x;a|q)=\left|\frac{\left({\mathrm% {e}^{2\mathrm{i}(\theta+\phi)}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm% {i}(\theta+2\phi)}},a{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|% ^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a{% \mathrm{e}^{\mathrm{i}\phi}})h(x,a{\mathrm{e}^{-\mathrm{i}\phi}})}}}} &

h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ ( θ + ϕ ) , α ⁢ e - i ⁢ ( θ + ϕ ) ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 italic-ϕ 𝛼 imaginary-unit 𝜃 italic-ϕ 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}(\theta+\phi)}},\alpha{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}};q% \right)_{\infty}}}} &

x = cos ⁡ ( θ + ϕ ) 𝑥 𝜃 italic-ϕ {\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}


Recurrence relation

2 ⁢ x ⁢ P n ⁡ ( x ; a | q ) = ( 1 - q n + 1 ) ⁢ P n + 1 ⁡ ( x ; a | q ) + 2 ⁢ a ⁢ q n ⁢ cos ⁡ ϕ ⁢ P n ⁡ ( x ; a | q ) + ( 1 - a 2 ⁢ q n - 1 ) ⁢ P n - 1 ⁡ ( x ; a | q ) 2 𝑥 q-Meixner-Pollaczek-polynomial-P 𝑛 𝑥 𝑎 𝑞 1 superscript 𝑞 𝑛 1 q-Meixner-Pollaczek-polynomial-P 𝑛 1 𝑥 𝑎 𝑞 2 𝑎 superscript 𝑞 𝑛 italic-ϕ q-Meixner-Pollaczek-polynomial-P 𝑛 𝑥 𝑎 𝑞 1 superscript 𝑎 2 superscript 𝑞 𝑛 1 q-Meixner-Pollaczek-polynomial-P 𝑛 1 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle 2xP_{n}\!\left(x;a|q\right)=(1-q^{n% +1})P_{n+1}\!\left(x;a|q\right)+2aq^{n}\cos\phi P_{n}\!\left(x;a|q\right){}+(1% -a^{2}q^{n-1})P_{n-1}\!\left(x;a|q\right)}}} {\displaystyle 2x\qMeixnerPollaczek{n}@{x}{a}{q}=(1-q^{n+1})\qMeixnerPollaczek{n+1}@{x}{a}{q}+2aq^n\cos@@{\phi} \qMeixnerPollaczek{n}@{x}{a}{q} {}+(1-a^2q^{n-1})\qMeixnerPollaczek{n-1}@{x}{a}{q} }

Monic recurrence relation

x ⁢ P ^ n ⁡ ( x ) = P ^ n + 1 ⁡ ( x ) + a ⁢ q n ⁢ cos ⁡ ϕ ⁢ P ^ n ⁡ ( x ) + 1 4 ⁢ ( 1 - q n ) ⁢ ( 1 - a 2 ⁢ q n - 1 ) ⁢ P ^ n - 1 ⁡ ( x ) 𝑥 q-Meixner-Pollaczek-polynomial-monic-p 𝑛 𝑥 𝑎 𝑞 q-Meixner-Pollaczek-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑞 𝑎 superscript 𝑞 𝑛 italic-ϕ q-Meixner-Pollaczek-polynomial-monic-p 𝑛 𝑥 𝑎 𝑞 1 4 1 superscript 𝑞 𝑛 1 superscript 𝑎 2 superscript 𝑞 𝑛 1 q-Meixner-Pollaczek-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}_{n}\!\left(x\right)=% {\widehat{P}}_{n+1}\!\left(x\right)+aq^{n}\cos\phi\,{\widehat{P}}_{n}\!\left(x% \right)+\frac{1}{4}(1-q^{n})(1-a^{2}q^{n-1}){\widehat{P}}_{n-1}\!\left(x\right% )}}} {\displaystyle x\monicqMeixnerPollaczek{n}@@{x}{a}{q}=\monicqMeixnerPollaczek{n+1}@@{x}{a}{q}+aq^n\cos@@{\phi}\,\monicqMeixnerPollaczek{n}@@{x}{a}{q}+\frac{1}{4}(1-q^n)(1-a^2q^{n-1})\monicqMeixnerPollaczek{n-1}@@{x}{a}{q} }
P n ⁡ ( x ; a | q ) = 2 n ( q ; q ) n ⁢ P ^ n ⁡ ( x ) q-Meixner-Pollaczek-polynomial-P 𝑛 𝑥 𝑎 𝑞 superscript 2 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Meixner-Pollaczek-polynomial-monic-p 𝑛 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a|q\right)=\frac{2^{% n}}{\left(q;q\right)_{n}}{\widehat{P}}_{n}\!\left(x\right)}}} {\displaystyle \qMeixnerPollaczek{n}@{x}{a}{q}=\frac{2^n}{\qPochhammer{q}{q}{n}}\monicqMeixnerPollaczek{n}@@{x}{a}{q} }

q-Difference equation

( 1 - q ) 2 ⁢ D q ⁢ [ w ~ ⁢ ( x ; a ⁢ q 1 2 | q ) ⁢ D q ⁢ y ⁢ ( x ) ] + 4 ⁢ q - n + 1 ⁢ ( 1 - q n ) ⁢ w ~ ⁢ ( x ; a | q ) ⁢ y ⁢ ( x ) = 0 superscript 1 𝑞 2 subscript 𝐷 𝑞 delimited-[] ~ 𝑤 𝑥 conditional 𝑎 superscript 𝑞 1 2 𝑞 subscript 𝐷 𝑞 𝑦 𝑥 4 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 ~ 𝑤 𝑥 conditional 𝑎 𝑞 𝑦 𝑥 0 {\displaystyle{\displaystyle{\displaystyle(1-q)^{2}D_{q}\left[{\tilde{w}}(x;aq% ^{\frac{1}{2}}|q)D_{q}y(x)\right]+4q^{-n+1}(1-q^{n}){\tilde{w}}(x;a|q)y(x)=0}}} {\displaystyle (1-q)^2D_q\left[{\tilde w}(x;aq^{\frac{1}{2}}|q)D_qy(x)\right] +4q^{-n+1}(1-q^n){\tilde w}(x;a|q)y(x)=0 }

Substitution(s): w ~ ⁢ ( x ; a | q ) := w ⁢ ( x ; a | q ) 1 - x 2 assign ~ 𝑤 𝑥 conditional 𝑎 𝑞 𝑤 𝑥 conditional 𝑎 𝑞 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a|q):=\frac{w(x;a|q)}{% \sqrt{1-x^{2}}}}}} &

y ⁢ ( x ) = P n ⁡ ( x ; a | q ) 𝑦 𝑥 q-Meixner-Pollaczek-polynomial-P 𝑛 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=P_{n}\!\left(x;a|q\right)}}} &
w ⁢ ( x ; a | q ) = | ( e 2 ⁢ i ⁢ ( θ + ϕ ) ; q ) ∞ ( a ⁢ e i ⁢ ( θ + 2 ⁢ ϕ ) , a ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ⁢ e i ⁢ ϕ ) ⁢ h ⁢ ( x , a ⁢ e - i ⁢ ϕ ) 𝑤 𝑥 conditional 𝑎 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 italic-ϕ 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 2 italic-ϕ 𝑎 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 imaginary-unit italic-ϕ ℎ 𝑥 𝑎 imaginary-unit italic-ϕ {\displaystyle{\displaystyle{\displaystyle w(x;a|q)=\left|\frac{\left({\mathrm% {e}^{2\mathrm{i}(\theta+\phi)}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm% {i}(\theta+2\phi)}},a{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|% ^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a{% \mathrm{e}^{\mathrm{i}\phi}})h(x,a{\mathrm{e}^{-\mathrm{i}\phi}})}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ ( θ + ϕ ) , α ⁢ e - i ⁢ ( θ + ϕ ) ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 italic-ϕ 𝛼 imaginary-unit 𝜃 italic-ϕ 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}(\theta+\phi)}},\alpha{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}};q% \right)_{\infty}}}} &

x = cos ⁡ ( θ + ϕ ) 𝑥 𝜃 italic-ϕ {\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}


Forward shift operator

δ q ⁢ P n ⁡ ( x ; a | q ) = - q - 1 2 ⁢ n ⁢ ( e i ⁢ ( θ + ϕ ) - e - i ⁢ ( θ + ϕ ) ) ⁢ P n - 1 ⁡ ( x ; a ⁢ q 1 2 | q ) subscript 𝛿 𝑞 q-Meixner-Pollaczek-polynomial-P 𝑛 𝑥 𝑎 𝑞 superscript 𝑞 1 2 𝑛 imaginary-unit 𝜃 italic-ϕ imaginary-unit 𝜃 italic-ϕ q-Meixner-Pollaczek-polynomial-P 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}P_{n}\!\left(x;a|q\right)=% -q^{-\frac{1}{2}n}({\mathrm{e}^{\mathrm{i}(\theta+\phi)}}-{\mathrm{e}^{-% \mathrm{i}(\theta+\phi)}})P_{n-1}\!\left(x;aq^{\frac{1}{2}}|q\right)}}} {\displaystyle \delta_q\qMeixnerPollaczek{n}@{x}{a}{q}=-q^{-\frac{1}{2}n}(\expe^{\iunit(\theta+\phi)}-\expe^{-\iunit(\theta+\phi)}) \qMeixnerPollaczek{n-1}@{x}{aq^{\frac{1}{2}}}{q} }

Substitution(s): x = cos ⁡ ( θ + ϕ ) 𝑥 𝜃 italic-ϕ {\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}


D q ⁢ P n ⁡ ( x ; a | q ) = 2 ⁢ q - 1 2 ⁢ ( n - 1 ) 1 - q ⁢ P n - 1 ⁡ ( x ; a ⁢ q 1 2 | q ) subscript 𝐷 𝑞 q-Meixner-Pollaczek-polynomial-P 𝑛 𝑥 𝑎 𝑞 2 superscript 𝑞 1 2 𝑛 1 1 𝑞 q-Meixner-Pollaczek-polynomial-P 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle D_{q}P_{n}\!\left(x;a|q\right)=% \frac{2q^{-\frac{1}{2}(n-1)}}{1-q}P_{n-1}\!\left(x;aq^{\frac{1}{2}}|q\right)}}} {\displaystyle D_q\qMeixnerPollaczek{n}@{x}{a}{q}=\frac{2q^{-\frac{1}{2}(n-1)}}{1-q}\qMeixnerPollaczek{n-1}@{x}{aq^{\frac{1}{2}}}{q} }

Backward shift operator

δ q ⁢ [ w ~ ⁢ ( x ; a | q ) ⁢ P n ⁡ ( x ; a | q ) ] = q - 1 2 ⁢ ( n + 1 ) ⁢ ( 1 - q n + 1 ) ⁢ ( e i ⁢ θ - e - i ⁢ θ ) ⁢ w ~ ⁢ ( x ; a ⁢ q - 1 2 | q ) ⁢ P n + 1 ⁡ ( x ; a ⁢ q - 1 2 | q ) subscript 𝛿 𝑞 delimited-[] ~ 𝑤 𝑥 conditional 𝑎 𝑞 q-Meixner-Pollaczek-polynomial-P 𝑛 𝑥 𝑎 𝑞 superscript 𝑞 1 2 𝑛 1 1 superscript 𝑞 𝑛 1 imaginary-unit 𝜃 imaginary-unit 𝜃 ~ 𝑤 𝑥 conditional 𝑎 superscript 𝑞 1 2 𝑞 q-Meixner-Pollaczek-polynomial-P 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}\left[{\tilde{w}}(x;a|q)P_% {n}\!\left(x;a|q\right)\right]{}=q^{-\frac{1}{2}(n+1)}(1-q^{n+1})({\mathrm{e}^% {\mathrm{i}\theta}}-{\mathrm{e}^{-\mathrm{i}\theta}}){}{\tilde{w}}(x;aq^{-% \frac{1}{2}}|q)P_{n+1}\!\left(x;aq^{-\frac{1}{2}}|q\right)}}} {\displaystyle \delta_q\left[{\tilde w}(x;a|q)\qMeixnerPollaczek{n}@{x}{a}{q}\right] {}=q^{-\frac{1}{2}(n+1)}(1-q^{n+1})(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) {} {\tilde w}(x;aq^{-\frac{1}{2}}|q)\qMeixnerPollaczek{n+1}@{x}{aq^{-\frac{1}{2}}}{q} }

Substitution(s): w ~ ⁢ ( x ; a | q ) := w ⁢ ( x ; a | q ) 1 - x 2 assign ~ 𝑤 𝑥 conditional 𝑎 𝑞 𝑤 𝑥 conditional 𝑎 𝑞 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a|q):=\frac{w(x;a|q)}{% \sqrt{1-x^{2}}}}}} &

w ⁢ ( x ; a | q ) = | ( e 2 ⁢ i ⁢ ( θ + ϕ ) ; q ) ∞ ( a ⁢ e i ⁢ ( θ + 2 ⁢ ϕ ) , a ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ⁢ e i ⁢ ϕ ) ⁢ h ⁢ ( x , a ⁢ e - i ⁢ ϕ ) 𝑤 𝑥 conditional 𝑎 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 italic-ϕ 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 2 italic-ϕ 𝑎 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 imaginary-unit italic-ϕ ℎ 𝑥 𝑎 imaginary-unit italic-ϕ {\displaystyle{\displaystyle{\displaystyle w(x;a|q)=\left|\frac{\left({\mathrm% {e}^{2\mathrm{i}(\theta+\phi)}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm% {i}(\theta+2\phi)}},a{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|% ^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a{% \mathrm{e}^{\mathrm{i}\phi}})h(x,a{\mathrm{e}^{-\mathrm{i}\phi}})}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ ( θ + ϕ ) , α ⁢ e - i ⁢ ( θ + ϕ ) ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 italic-ϕ 𝛼 imaginary-unit 𝜃 italic-ϕ 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}(\theta+\phi)}},\alpha{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}};q% \right)_{\infty}}}} &

x = cos ⁡ ( θ + ϕ ) 𝑥 𝜃 italic-ϕ {\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}


D q ⁢ [ w ~ ⁢ ( x ; a | q ) ⁢ P n ⁡ ( x ; a | q ) ] = - 2 ⁢ q - 1 2 ⁢ n ⁢ 1 - q n + 1 1 - q ⁢ w ~ ⁢ ( x ; a ⁢ q - 1 2 | q ) ⁢ P n + 1 ⁡ ( x ; a ⁢ q - 1 2 | q ) subscript 𝐷 𝑞 delimited-[] ~ 𝑤 𝑥 conditional 𝑎 𝑞 q-Meixner-Pollaczek-polynomial-P 𝑛 𝑥 𝑎 𝑞 2 superscript 𝑞 1 2 𝑛 1 superscript 𝑞 𝑛 1 1 𝑞 ~ 𝑤 𝑥 conditional 𝑎 superscript 𝑞 1 2 𝑞 q-Meixner-Pollaczek-polynomial-P 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle D_{q}\left[{\tilde{w}}(x;a|q)P_{n}% \!\left(x;a|q\right)\right]=-2q^{-\frac{1}{2}n}\frac{1-q^{n+1}}{1-q}{\tilde{w}% }(x;aq^{-\frac{1}{2}}|q)P_{n+1}\!\left(x;aq^{-\frac{1}{2}}|q\right)}}} {\displaystyle D_q\left[{\tilde w}(x;a|q)\qMeixnerPollaczek{n}@{x}{a}{q}\right]= -2q^{-\frac{1}{2}n}\frac{1-q^{n+1}}{1-q}{\tilde w}(x;aq^{-\frac{1}{2}}|q)\qMeixnerPollaczek{n+1}@{x}{aq^{-\frac{1}{2}}}{q} }

Substitution(s): w ~ ⁢ ( x ; a | q ) := w ⁢ ( x ; a | q ) 1 - x 2 assign ~ 𝑤 𝑥 conditional 𝑎 𝑞 𝑤 𝑥 conditional 𝑎 𝑞 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a|q):=\frac{w(x;a|q)}{% \sqrt{1-x^{2}}}}}} &

w ⁢ ( x ; a | q ) = | ( e 2 ⁢ i ⁢ ( θ + ϕ ) ; q ) ∞ ( a ⁢ e i ⁢ ( θ + 2 ⁢ ϕ ) , a ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ⁢ e i ⁢ ϕ ) ⁢ h ⁢ ( x , a ⁢ e - i ⁢ ϕ ) 𝑤 𝑥 conditional 𝑎 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 italic-ϕ 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 2 italic-ϕ 𝑎 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 imaginary-unit italic-ϕ ℎ 𝑥 𝑎 imaginary-unit italic-ϕ {\displaystyle{\displaystyle{\displaystyle w(x;a|q)=\left|\frac{\left({\mathrm% {e}^{2\mathrm{i}(\theta+\phi)}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm% {i}(\theta+2\phi)}},a{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|% ^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a{% \mathrm{e}^{\mathrm{i}\phi}})h(x,a{\mathrm{e}^{-\mathrm{i}\phi}})}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ ( θ + ϕ ) , α ⁢ e - i ⁢ ( θ + ϕ ) ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 italic-ϕ 𝛼 imaginary-unit 𝜃 italic-ϕ 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}(\theta+\phi)}},\alpha{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}};q% \right)_{\infty}}}} &

x = cos ⁡ ( θ + ϕ ) 𝑥 𝜃 italic-ϕ {\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}


Rodrigues-type formula

w ~ ⁢ ( x ; a | q ) ⁢ P n ⁡ ( x ; a | q ) = ( q - 1 2 ) n ⁢ q 1 4 ⁢ n ⁢ ( n - 1 ) ⁢ 1 ( q ; q ) n ⁢ ( D q ) n ⁢ [ w ~ ⁢ ( x ; a ⁢ q 1 2 ⁢ n | q ) ] ~ 𝑤 𝑥 conditional 𝑎 𝑞 q-Meixner-Pollaczek-polynomial-P 𝑛 𝑥 𝑎 𝑞 superscript 𝑞 1 2 𝑛 superscript 𝑞 1 4 𝑛 𝑛 1 1 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript subscript 𝐷 𝑞 𝑛 delimited-[] ~ 𝑤 𝑥 conditional 𝑎 superscript 𝑞 1 2 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a|q)P_{n}\!\left(x;a|q% \right)=\left(\frac{q-1}{2}\right)^{n}q^{\frac{1}{4}n(n-1)}\frac{1}{\left(q;q% \right)_{n}}\left(D_{q}\right)^{n}\left[{\tilde{w}}(x;aq^{\frac{1}{2}n}|q)% \right]}}} {\displaystyle {\tilde w}(x;a|q)\qMeixnerPollaczek{n}@{x}{a}{q}=\left(\frac{q-1}{2}\right)^n q^{\frac{1}{4}n(n-1)}\frac{1}{\qPochhammer{q}{q}{n}}\left(D_q\right)^n\left[{\tilde w}(x;aq^{\frac{1}{2}n}|q)\right] }

Substitution(s): w ~ ⁢ ( x ; a | q ) := w ⁢ ( x ; a | q ) 1 - x 2 assign ~ 𝑤 𝑥 conditional 𝑎 𝑞 𝑤 𝑥 conditional 𝑎 𝑞 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a|q):=\frac{w(x;a|q)}{% \sqrt{1-x^{2}}}}}} &

w ⁢ ( x ; a | q ) = | ( e 2 ⁢ i ⁢ ( θ + ϕ ) ; q ) ∞ ( a ⁢ e i ⁢ ( θ + 2 ⁢ ϕ ) , a ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ⁢ e i ⁢ ϕ ) ⁢ h ⁢ ( x , a ⁢ e - i ⁢ ϕ ) 𝑤 𝑥 conditional 𝑎 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 italic-ϕ 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 2 italic-ϕ 𝑎 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 imaginary-unit italic-ϕ ℎ 𝑥 𝑎 imaginary-unit italic-ϕ {\displaystyle{\displaystyle{\displaystyle w(x;a|q)=\left|\frac{\left({\mathrm% {e}^{2\mathrm{i}(\theta+\phi)}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm% {i}(\theta+2\phi)}},a{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|% ^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a{% \mathrm{e}^{\mathrm{i}\phi}})h(x,a{\mathrm{e}^{-\mathrm{i}\phi}})}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ ( θ + ϕ ) , α ⁢ e - i ⁢ ( θ + ϕ ) ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 italic-ϕ 𝛼 imaginary-unit 𝜃 italic-ϕ 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}(\theta+\phi)}},\alpha{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}};q% \right)_{\infty}}}} &

x = cos ⁡ ( θ + ϕ ) 𝑥 𝜃 italic-ϕ {\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}


Generating functions

| ( a ⁢ e i ⁢ ϕ ⁢ t ; q ) ∞ ( e i ⁢ ( θ + ϕ ) ⁢ t ; q ) ∞ | 2 = ( a ⁢ e i ⁢ ϕ ⁢ t , a ⁢ e - i ⁢ ϕ ⁢ t ; q ) ∞ ( e i ⁢ ( θ + ϕ ) ⁢ t , e - i ⁢ ( θ + ϕ ) ⁢ t ; q ) ∞ superscript q-Pochhammer-symbol 𝑎 imaginary-unit italic-ϕ 𝑡 𝑞 q-Pochhammer-symbol imaginary-unit 𝜃 italic-ϕ 𝑡 𝑞 2 q-Pochhammer-symbol 𝑎 imaginary-unit italic-ϕ 𝑡 𝑎 imaginary-unit italic-ϕ 𝑡 𝑞 q-Pochhammer-symbol imaginary-unit 𝜃 italic-ϕ 𝑡 imaginary-unit 𝜃 italic-ϕ 𝑡 𝑞 {\displaystyle{\displaystyle{\displaystyle\left|\frac{\left(a{\mathrm{e}^{% \mathrm{i}\phi}}t;q\right)_{\infty}}{\left({\mathrm{e}^{\mathrm{i}(\theta+\phi% )}}t;q\right)_{\infty}}\right|^{2}{}=\frac{\left(a{\mathrm{e}^{\mathrm{i}\phi}% }t,a{\mathrm{e}^{-\mathrm{i}\phi}}t;q\right)_{\infty}}{\left({\mathrm{e}^{% \mathrm{i}(\theta+\phi)}}t,{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}}t;q\right)_{% \infty}}}}} {\displaystyle \left|\frac{\qPochhammer{a\expe^{\iunit\phi}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit(\theta+\phi)}t}{q}{\infty}}\right|^2 {}=\frac{\qPochhammer{a\expe^{\iunit\phi}t,a\expe^{-\iunit\phi}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit(\theta+\phi)}t,\expe^{-\iunit(\theta+\phi)}t}{q}{\infty}} }

Substitution(s): x = cos ⁡ ( θ + ϕ ) 𝑥 𝜃 italic-ϕ {\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}


| ( a ⁢ e i ⁢ ϕ ⁢ t ; q ) ∞ ( e i ⁢ ( θ + ϕ ) ⁢ t ; q ) ∞ | 2 = ∑ n = 0 ∞ P n ⁡ ( x ; a | q ) ⁢ t n superscript q-Pochhammer-symbol 𝑎 imaginary-unit italic-ϕ 𝑡 𝑞 q-Pochhammer-symbol imaginary-unit 𝜃 italic-ϕ 𝑡 𝑞 2 superscript subscript 𝑛 0 q-Meixner-Pollaczek-polynomial-P 𝑛 𝑥 𝑎 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left|\frac{\left(a{\mathrm{e}^{% \mathrm{i}\phi}}t;q\right)_{\infty}}{\left({\mathrm{e}^{\mathrm{i}(\theta+\phi% )}}t;q\right)_{\infty}}\right|^{2}{}=\sum_{n=0}^{\infty}P_{n}\!\left(x;a|q% \right)t^{n}}}} {\displaystyle \left|\frac{\qPochhammer{a\expe^{\iunit\phi}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit(\theta+\phi)}t}{q}{\infty}}\right|^2 {}=\sum_{n=0}^{\infty}\qMeixnerPollaczek{n}@{x}{a}{q}t^n }
1 ( e i ⁢ ( θ + ϕ ) ⁢ t ; q ) ∞ ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ a ⁢ e i ⁢ ( θ + 2 ⁢ ϕ ) , a ⁢ e i ⁢ θ ⁢ a 2 ⁢ q ⁢ e - i ⁢ ( θ + ϕ ) ⁢ t = ∑ n = 0 ∞ P n ⁡ ( x ; a | q ) ( a 2 ; q ) n ⁢ t n 1 q-Pochhammer-symbol imaginary-unit 𝜃 italic-ϕ 𝑡 𝑞 \qHyperrphis 21 @ @ 𝑎 imaginary-unit 𝜃 2 italic-ϕ 𝑎 imaginary-unit 𝜃 superscript 𝑎 2 𝑞 imaginary-unit 𝜃 italic-ϕ 𝑡 superscript subscript 𝑛 0 q-Meixner-Pollaczek-polynomial-P 𝑛 𝑥 𝑎 𝑞 q-Pochhammer-symbol superscript 𝑎 2 𝑞 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{\left({\mathrm{e}^{\mathrm{% i}(\theta+\phi)}}t;q\right)_{\infty}}\ \qHyperrphis{2}{1}@@{a{\mathrm{e}^{% \mathrm{i}(\theta+2\phi)}},a{\mathrm{e}^{\mathrm{i}\theta}}}{a^{2}}{q}{{% \mathrm{e}^{-\mathrm{i}(\theta+\phi)}}t}{}=\sum_{n=0}^{\infty}\frac{P_{n}\!% \left(x;a|q\right)}{\left(a^{2};q\right)_{n}}t^{n}}}} {\displaystyle \frac{1}{\qPochhammer{\expe^{\iunit(\theta+\phi)}t}{q}{\infty}}\ \qHyperrphis{2}{1}@@{a\expe^{\iunit(\theta+2\phi)},a\expe^{\iunit\theta}}{a^2}{q}{\expe^{-\iunit(\theta+\phi)}t} {}=\sum_{n=0}^{\infty}\frac{\qMeixnerPollaczek{n}@{x}{a}{q}}{\qPochhammer{a^2}{q}{n}}t^n }

Substitution(s): x = cos ⁡ ( θ + ϕ ) 𝑥 𝜃 italic-ϕ {\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}


Limit relations

Continuous q-Hahn polynomial to q-Meixner-Pollaczek polynomial

p n ⁡ ( cos ⁡ ( θ + ϕ ) ; a , 0 , 0 , a ; q ) ( q ; q ) n = P n ⁡ ( cos ⁡ ( θ + ϕ ) ; a | q ) continuous-q-Hahn-polynomial-p 𝑛 𝜃 italic-ϕ 𝑎 0 0 𝑎 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Meixner-Pollaczek-polynomial-P 𝑛 𝜃 italic-ϕ 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{p_{n}\!\left(\cos\left(\theta+% \phi\right);a,0,0,a;q\right)}{\left(q;q\right)_{n}}=P_{n}\!\left(\cos\left(% \theta+\phi\right);a|q\right)}}} {\displaystyle \frac{\ctsqHahn{n}@{\cos@{\theta+\phi}}{a}{0}{0}{a}{q}}{\qPochhammer{q}{q}{n}}=\qMeixnerPollaczek{n}@{\cos@{\theta+\phi}}{a}{q} }

q-Meixner-Pollaczek polynomial to Continuous q-ultraspherical /

P n ⁡ ( cos ⁡ ϕ ; β | q ) = C n ⁡ ( cos ⁡ ϕ ; β | q ) q-Meixner-Pollaczek-polynomial-P 𝑛 italic-ϕ 𝛽 𝑞 continuous-q-ultraspherical-Rogers-polynomial 𝑛 italic-ϕ 𝛽 𝑞 {\displaystyle{\displaystyle{\displaystyle\par P_{n}\!\left(\cos\phi;\beta|q% \right)=C_{n}\!\left(\cos\phi;\beta\,|\,q\right)}}} {\displaystyle \qMeixnerPollaczek{n}@{\cos@@{\phi}}{\beta}{q}=\ctsqUltra{n}@{\cos@@{\phi}}{\beta}{q} }

q-Meixner-Pollaczek polynomial to Continuous

P n ⁡ ( cos ⁡ ( θ + ϕ ) ; q 1 2 ⁢ α + 1 2 | q ) = q - ( 1 2 ⁢ α + 1 4 ) ⁢ n ⁢ P n ( α ) ⁡ ( cos ⁡ θ | q ) q-Meixner-Pollaczek-polynomial-P 𝑛 𝜃 italic-ϕ superscript 𝑞 1 2 𝛼 1 2 𝑞 superscript 𝑞 1 2 𝛼 1 4 𝑛 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝜃 𝑞 {\displaystyle{\displaystyle{\displaystyle\par P_{n}\!\left(\cos\left(\theta+% \phi\right);q^{\frac{1}{2}\alpha+\frac{1}{2}}|q\right)=q^{-(\frac{1}{2}\alpha+% \frac{1}{4})n}P^{(\alpha)}_{n}\!\left(\cos\theta|q\right)}}} {\displaystyle \qMeixnerPollaczek{n}@{\cos@{\theta+\phi}}{q^{\frac{1}{2}\alpha+\frac{1}{2}}}{q}= q^{-(\frac{1}{2}\alpha+\frac{1}{4})n}\ctsqLaguerre{\alpha}{n}@{\cos@@{\theta}}{q} }

q-Meixner-Pollaczek polynomial to Meixner-Pollaczek polynomial

lim q → 1 ⁡ P n ⁡ ( cos ⁡ ( ln ⁡ q - x + ϕ ) ; q λ | q ) = P n ( λ ) ⁡ ( x ; - ϕ ) subscript → 𝑞 1 q-Meixner-Pollaczek-polynomial-P 𝑛 superscript 𝑞 𝑥 italic-ϕ superscript 𝑞 𝜆 𝑞 Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}P_{n}\!\left(% \cos\left(\ln q^{-x}+\phi\right);q^{\lambda}|q\right)=P^{(\lambda)}_{n}\!\left% (x;-\phi\right)}}} {\displaystyle \lim_{q\rightarrow 1}\qMeixnerPollaczek{n}@{\cos@{\ln@@{q^{-x}}+\phi}}{q^{\lambda}}{q} =\MeixnerPollaczek{\lambda}{n}@{x}{-\phi} }

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