Continuous q-Laguerre

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Continuous q-Laguerre

Basic hypergeometric representation

P n ( α ) ⁡ ( x | q ) = ( q α + 1 ; q ) n ( q ; q ) n ⁢ \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ q - n , q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ , q 1 2 ⁢ α + 1 4 ⁢ e - i ⁢ θ ⁢ q α + 1 , 0 ⁢ q ⁢ q continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 𝛼 1 0 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha)}_{n}\!\left(x|q\right)=% \frac{\left(q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}}\,\qHyperrphis{3}{% 2}@@{q^{-n},q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},q% ^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{-\mathrm{i}\theta}}}{q^{\alpha+1}% ,0}{q}{q}}}} {\displaystyle \ctsqLaguerre{\alpha}{n}@{x}{q}=\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}}\,\qHyperrphis{3}{2}@@{q^{-n},q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta},q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{-\iunit\theta}}{q^{\alpha+1},0}{q}{q} }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


P n ( α ) ⁡ ( x | q ) = ( q 1 2 ⁢ α + 3 4 ⁢ e - i ⁢ θ ; q ) n ( q ; q ) n ⁢ q ( 1 2 ⁢ α + 1 4 ) ⁢ n ⁢ e i ⁢ n ⁢ θ ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - n , q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ ⁢ q - 1 2 ⁢ α + 1 4 - n ⁢ e i ⁢ θ ⁢ q ⁢ q - 1 2 ⁢ α + 1 4 ⁢ e - i ⁢ θ continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 q-Pochhammer-symbol superscript 𝑞 1 2 𝛼 3 4 imaginary-unit 𝜃 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 1 2 𝛼 1 4 𝑛 imaginary-unit 𝑛 𝜃 \qHyperrphis 21 @ @ superscript 𝑞 𝑛 superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛼 1 4 𝑛 imaginary-unit 𝜃 𝑞 superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha)}_{n}\!\left(x|q\right)=% \frac{\left(q^{\frac{1}{2}\alpha+\frac{3}{4}}{\mathrm{e}^{-\mathrm{i}\theta}};% q\right)_{n}}{\left(q;q\right)_{n}}q^{(\frac{1}{2}\alpha+\frac{1}{4})n}{% \mathrm{e}^{\mathrm{i}n\theta}}{}\qHyperrphis{2}{1}@@{q^{-n},q^{\frac{1}{2}% \alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}}{q^{-\frac{1}{2}\alpha+% \frac{1}{4}-n}{\mathrm{e}^{\mathrm{i}\theta}}}{q}{q^{-\frac{1}{2}\alpha+\frac{% 1}{4}}{\mathrm{e}^{-\mathrm{i}\theta}}}}}} {\displaystyle \ctsqLaguerre{\alpha}{n}@{x}{q}=\frac{\qPochhammer{q^{\frac{1}{2}\alpha+\frac{3}{4}}\expe^{-\iunit\theta}}{q}{n}}{\qPochhammer{q}{q}{n}} q^{(\frac{1}{2}\alpha+\frac{1}{4})n}\expe^{\iunit n\theta} {}\qHyperrphis{2}{1}@@{q^{-n},q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta}} {q^{-\frac{1}{2}\alpha+\frac{1}{4}-n}\expe^{\iunit\theta}} {q}{q^{-\frac{1}{2}\alpha+\frac{1}{4}}\expe^{-\iunit\theta}} }

Orthogonality relation(s)

1 2 ⁢ π ⁢ ∫ - 1 1 w ⁢ ( x ) 1 - x 2 ⁢ P m ( α ) ⁡ ( x | q ) ⁢ P n ( α ) ⁡ ( x | q ) ⁢ 𝑑 x = 1 ( q , q α + 1 ; q ) ∞ ⁢ ( q α + 1 ; q ) n ( q ; q ) n ⁢ q ( α + 1 2 ) ⁢ n ⁢ δ m , n 1 2 superscript subscript 1 1 𝑤 𝑥 1 superscript 𝑥 2 continuous-q-Laguerre-polynomial-P 𝛼 𝑚 𝑥 𝑞 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 differential-d 𝑥 1 q-Pochhammer-symbol 𝑞 superscript 𝑞 𝛼 1 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 𝛼 1 2 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x% )}{\sqrt{1-x^{2}}}P^{(\alpha)}_{m}\!\left(x|q\right)P^{(\alpha)}_{n}\!\left(x|% q\right)\,dx{}=\frac{1}{\left(q,q^{\alpha+1};q\right)_{\infty}}\frac{\left(q^{% \alpha+1};q\right)_{n}}{\left(q;q\right)_{n}}q^{(\alpha+\frac{1}{2})n}\,\delta% _{m,n}}}} {\displaystyle \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\ctsqLaguerre{\alpha}{m}@{x}{q}\ctsqLaguerre{\alpha}{n}@{x}{q}\,dx {}=\frac{1}{\qPochhammer{q,q^{\alpha+1}}{q}{\infty}}\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}} q^{(\alpha+\frac{1}{2})n}\,\Kronecker{m}{n} }

Substitution(s): w ⁢ ( x ) := w ⁢ ( x ; q α | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ ⁢ q 1 2 ⁢ α + 3 4 ⁢ e i ⁢ θ ; q ) ∞ | 2 = | ( e i ⁢ θ , - e i ⁢ θ ; q 1 2 ) ∞ ( q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ ; q 1 2 ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , q 1 2 ⁢ α + 1 4 ) ⁢ h ⁢ ( x , q 1 2 ⁢ α + 3 4 ) assign 𝑤 𝑥 𝑤 𝑥 conditional superscript 𝑞 𝛼 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛼 3 4 imaginary-unit 𝜃 𝑞 2 superscript q-Pochhammer-symbol imaginary-unit 𝜃 imaginary-unit 𝜃 superscript 𝑞 1 2 q-Pochhammer-symbol superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 𝛼 1 4 ℎ 𝑥 superscript 𝑞 1 2 𝛼 3 4 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha}|q)=\left|\frac% {\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(q^{\frac{1}{2% }\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}q^{\frac{1}{2}\alpha+\frac{% 3}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\left|% \frac{\left({\mathrm{e}^{\mathrm{i}\theta}},-{\mathrm{e}^{\mathrm{i}\theta}};q% ^{\frac{1}{2}}\right)_{\infty}}{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}{% \mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\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,q^{\frac{1}{% 2}\alpha+\frac{1}{4}})h(x,q^{\frac{1}{2}\alpha+\frac{3}{4}})}}}} &

h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; 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 𝛼 imaginary-unit 𝜃 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}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


Recurrence relation

2 ⁢ x ⁢ P n ( α ) ⁡ ( x | q ) = q - 1 2 ⁢ α - 1 4 ⁢ ( 1 - q n + 1 ) ⁢ P n + 1 ( α ) ⁡ ( x | q ) + q n + 1 2 ⁢ α + 1 4 ⁢ ( 1 + q 1 2 ) ⁢ P n ( α ) ⁡ ( x | q ) + q 1 2 ⁢ α + 1 4 ⁢ ( 1 - q n + α ) ⁢ P n - 1 ( α ) ⁡ ( x | q ) 2 𝑥 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 superscript 𝑞 1 2 𝛼 1 4 1 superscript 𝑞 𝑛 1 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 1 𝑥 𝑞 superscript 𝑞 𝑛 1 2 𝛼 1 4 1 superscript 𝑞 1 2 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 superscript 𝑞 1 2 𝛼 1 4 1 superscript 𝑞 𝑛 𝛼 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle 2xP^{(\alpha)}_{n}\!\left(x|q\right% )=q^{-\frac{1}{2}\alpha-\frac{1}{4}}(1-q^{n+1})P^{(\alpha)}_{n+1}\!\left(x|q% \right){}+q^{n+\frac{1}{2}\alpha+\frac{1}{4}}(1+q^{\frac{1}{2}})P^{(\alpha)}_{% n}\!\left(x|q\right){}+q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^{n+\alpha})P^{(% \alpha)}_{n-1}\!\left(x|q\right)}}} {\displaystyle 2x\ctsqLaguerre{\alpha}{n}@{x}{q}=q^{-\frac{1}{2}\alpha-\frac{1}{4}}(1-q^{n+1})\ctsqLaguerre{\alpha}{n+1}@{x}{q} {}+q^{n+\frac{1}{2}\alpha+\frac{1}{4}}(1+q^{\frac{1}{2}})\ctsqLaguerre{\alpha}{n}@{x}{q} {}+q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^{n+\alpha})\ctsqLaguerre{\alpha}{n-1}@{x}{q} }

Monic recurrence relation

x ⁢ P ^ n ( α ) ⁡ ( x ) = P ^ n + 1 ( α ) ⁡ ( x ) + 1 2 ⁢ q n + 1 2 ⁢ α + 1 4 ⁢ ( 1 + q 1 2 ) ⁢ P ^ n ( α ) ⁡ ( x ) + 1 4 ⁢ ( 1 - q n ) ⁢ ( 1 - q n + α ) ⁢ P ^ n - 1 ( α ) ⁡ ( x ) 𝑥 continuous-q-Laguerre-polynomial-monic-p 𝛼 𝑛 𝑥 𝑞 continuous-q-Laguerre-polynomial-monic-p 𝛼 𝑛 1 𝑥 𝑞 1 2 superscript 𝑞 𝑛 1 2 𝛼 1 4 1 superscript 𝑞 1 2 continuous-q-Laguerre-polynomial-monic-p 𝛼 𝑛 𝑥 𝑞 1 4 1 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 𝛼 continuous-q-Laguerre-polynomial-monic-p 𝛼 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}^{(\alpha)}_{n}\!% \left(x\right)={\widehat{P}}^{(\alpha)}_{n+1}\!\left(x\right)+\frac{1}{2}q^{n+% \frac{1}{2}\alpha+\frac{1}{4}}(1+q^{\frac{1}{2}}){\widehat{P}}^{(\alpha)}_{n}% \!\left(x\right){}+\frac{1}{4}(1-q^{n})(1-q^{n+\alpha}){\widehat{P}}^{(\alpha)% }_{n-1}\!\left(x\right)}}} {\displaystyle x\monicctsqLaguerre{\alpha}{n}@@{x}{q}=\monicctsqLaguerre{\alpha}{n+1}@@{x}{q}+\frac{1}{2}q^{n+\frac{1}{2}\alpha+\frac{1}{4}}(1+q^{\frac{1}{2}})\monicctsqLaguerre{\alpha}{n}@@{x}{q} {}+\frac{1}{4}(1-q^n)(1-q^{n+\alpha})\monicctsqLaguerre{\alpha}{n-1}@@{x}{q} }
P n ( α ) ⁡ ( x | q ) = 2 n ⁢ q ( 1 2 ⁢ α + 1 4 ) ⁢ n ( q ; q ) n ⁢ P ^ n ( α ) ⁡ ( x ) continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 superscript 2 𝑛 superscript 𝑞 1 2 𝛼 1 4 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 continuous-q-Laguerre-polynomial-monic-p 𝛼 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha)}_{n}\!\left(x|q\right)=% \frac{2^{n}q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}{\left(q;q\right)_{n}}{% \widehat{P}}^{(\alpha)}_{n}\!\left(x\right)}}} {\displaystyle \ctsqLaguerre{\alpha}{n}@{x}{q}=\frac{2^nq^{(\frac{1}{2}\alpha+\frac{1}{4})n}}{\qPochhammer{q}{q}{n}}\monicctsqLaguerre{\alpha}{n}@@{x}{q} }

q-Difference equations

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

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

y ⁢ ( x ) = P n ( α ) ⁡ ( x | q ) 𝑦 𝑥 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=P^{(\alpha)}_{n}\!\left(x|q% \right)}}} &
w ⁢ ( x ) := w ⁢ ( x ; q α | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ ⁢ q 1 2 ⁢ α + 3 4 ⁢ e i ⁢ θ ; q ) ∞ | 2 = | ( e i ⁢ θ , - e i ⁢ θ ; q 1 2 ) ∞ ( q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ ; q 1 2 ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , q 1 2 ⁢ α + 1 4 ) ⁢ h ⁢ ( x , q 1 2 ⁢ α + 3 4 ) assign 𝑤 𝑥 𝑤 𝑥 conditional superscript 𝑞 𝛼 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛼 3 4 imaginary-unit 𝜃 𝑞 2 superscript q-Pochhammer-symbol imaginary-unit 𝜃 imaginary-unit 𝜃 superscript 𝑞 1 2 q-Pochhammer-symbol superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 𝛼 1 4 ℎ 𝑥 superscript 𝑞 1 2 𝛼 3 4 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha}|q)=\left|\frac% {\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(q^{\frac{1}{2% }\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}q^{\frac{1}{2}\alpha+\frac{% 3}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\left|% \frac{\left({\mathrm{e}^{\mathrm{i}\theta}},-{\mathrm{e}^{\mathrm{i}\theta}};q% ^{\frac{1}{2}}\right)_{\infty}}{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}{% \mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\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,q^{\frac{1}{% 2}\alpha+\frac{1}{4}})h(x,q^{\frac{1}{2}\alpha+\frac{3}{4}})}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; 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 𝛼 imaginary-unit 𝜃 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}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


Forward shift operator

δ q ⁢ P n ( α ) ⁡ ( x | q ) = - q - n + 1 2 ⁢ α + 3 4 ⁢ ( e i ⁢ θ - e - i ⁢ θ ) ⁢ P n - 1 ( α + 1 ) ⁡ ( x | q ) subscript 𝛿 𝑞 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 superscript 𝑞 𝑛 1 2 𝛼 3 4 imaginary-unit 𝜃 imaginary-unit 𝜃 continuous-q-Laguerre-polynomial-P 𝛼 1 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}P^{(\alpha)}_{n}\!\left(x|% q\right)=-q^{-n+\frac{1}{2}\alpha+\frac{3}{4}}({\mathrm{e}^{\mathrm{i}\theta}}% -{\mathrm{e}^{-\mathrm{i}\theta}})P^{(\alpha+1)}_{n-1}\!\left(x|q\right)}}} {\displaystyle \delta_q\ctsqLaguerre{\alpha}{n}@{x}{q}=-q^{-n+\frac{1}{2}\alpha+\frac{3}{4}} (\expe^{\iunit\theta}-\expe^{-\iunit\theta})\ctsqLaguerre{\alpha+1}{n-1}@{x}{q} }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


D q ⁢ P n ( α ) ⁡ ( x | q ) = 2 ⁢ q - n + 1 2 ⁢ α + 5 4 1 - q ⁢ P n - 1 ( α + 1 ) ⁡ ( x | q ) subscript 𝐷 𝑞 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 2 superscript 𝑞 𝑛 1 2 𝛼 5 4 1 𝑞 continuous-q-Laguerre-polynomial-P 𝛼 1 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle D_{q}P^{(\alpha)}_{n}\!\left(x|q% \right)=\frac{2q^{-n+\frac{1}{2}\alpha+\frac{5}{4}}}{1-q}P^{(\alpha+1)}_{n-1}% \!\left(x|q\right)}}} {\displaystyle D_q\ctsqLaguerre{\alpha}{n}@{x}{q}=\frac{2q^{-n+\frac{1}{2}\alpha+\frac{5}{4}}}{1-q} \ctsqLaguerre{\alpha+1}{n-1}@{x}{q} }

Backward shift operator

δ q ⁢ [ w ~ ⁢ ( x ; q α | q ) ⁢ P n ( α ) ⁡ ( x | q ) ] = q - 1 2 ⁢ α - 1 4 ⁢ ( 1 - q n + 1 ) ⁢ ( e i ⁢ θ - e - i ⁢ θ ) ⁢ w ~ ⁢ ( x ; q α - 1 | q ) ⁢ P n + 1 ( α - 1 ) ⁡ ( x | q ) subscript 𝛿 𝑞 delimited-[] ~ 𝑤 𝑥 conditional superscript 𝑞 𝛼 𝑞 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 superscript 𝑞 1 2 𝛼 1 4 1 superscript 𝑞 𝑛 1 imaginary-unit 𝜃 imaginary-unit 𝜃 ~ 𝑤 𝑥 conditional superscript 𝑞 𝛼 1 𝑞 continuous-q-Laguerre-polynomial-P 𝛼 1 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}\left[{\tilde{w}}(x;q^{% \alpha}|q)P^{(\alpha)}_{n}\!\left(x|q\right)\right]{}=q^{-\frac{1}{2}\alpha-% \frac{1}{4}}(1-q^{n+1})({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e}^{-\mathrm{% i}\theta}}){}{\tilde{w}}(x;q^{\alpha-1}|q)P^{(\alpha-1)}_{n+1}\!\left(x|q% \right)}}} {\displaystyle \delta_q\left[{\tilde w}(x;q^{\alpha}|q)\ctsqLaguerre{\alpha}{n}@{x}{q}\right] {}=q^{-\frac{1}{2}\alpha-\frac{1}{4}}(1-q^{n+1})(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) {} {\tilde w}(x;q^{\alpha-1}|q)\ctsqLaguerre{\alpha-1}{n+1}@{x}{q} }

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

w ⁢ ( x ) := w ⁢ ( x ; q α | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ ⁢ q 1 2 ⁢ α + 3 4 ⁢ e i ⁢ θ ; q ) ∞ | 2 = | ( e i ⁢ θ , - e i ⁢ θ ; q 1 2 ) ∞ ( q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ ; q 1 2 ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , q 1 2 ⁢ α + 1 4 ) ⁢ h ⁢ ( x , q 1 2 ⁢ α + 3 4 ) assign 𝑤 𝑥 𝑤 𝑥 conditional superscript 𝑞 𝛼 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛼 3 4 imaginary-unit 𝜃 𝑞 2 superscript q-Pochhammer-symbol imaginary-unit 𝜃 imaginary-unit 𝜃 superscript 𝑞 1 2 q-Pochhammer-symbol superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 𝛼 1 4 ℎ 𝑥 superscript 𝑞 1 2 𝛼 3 4 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha}|q)=\left|\frac% {\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(q^{\frac{1}{2% }\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}q^{\frac{1}{2}\alpha+\frac{% 3}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\left|% \frac{\left({\mathrm{e}^{\mathrm{i}\theta}},-{\mathrm{e}^{\mathrm{i}\theta}};q% ^{\frac{1}{2}}\right)_{\infty}}{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}{% \mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\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,q^{\frac{1}{% 2}\alpha+\frac{1}{4}})h(x,q^{\frac{1}{2}\alpha+\frac{3}{4}})}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; 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 𝛼 imaginary-unit 𝜃 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}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


D q ⁢ [ w ~ ⁢ ( x ; q α | q ) ⁢ P n ( α ) ⁡ ( x | q ) ] = - 2 ⁢ q - 1 2 ⁢ α + 1 4 ⁢ 1 - q n + 1 1 - q ⁢ w ~ ⁢ ( x ; q α - 1 | q ) ⁢ P n + 1 ( α - 1 ) ⁡ ( x | q ) subscript 𝐷 𝑞 delimited-[] ~ 𝑤 𝑥 conditional superscript 𝑞 𝛼 𝑞 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 2 superscript 𝑞 1 2 𝛼 1 4 1 superscript 𝑞 𝑛 1 1 𝑞 ~ 𝑤 𝑥 conditional superscript 𝑞 𝛼 1 𝑞 continuous-q-Laguerre-polynomial-P 𝛼 1 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle D_{q}\left[{\tilde{w}}(x;q^{\alpha}% |q)P^{(\alpha)}_{n}\!\left(x|q\right)\right]{}=-2q^{-\frac{1}{2}\alpha+\frac{1% }{4}}\frac{1-q^{n+1}}{1-q}{\tilde{w}}(x;q^{\alpha-1}|q)P^{(\alpha-1)}_{n+1}\!% \left(x|q\right)}}} {\displaystyle D_q\left[{\tilde w}(x;q^{\alpha}|q)\ctsqLaguerre{\alpha}{n}@{x}{q}\right] {}=-2q^{-\frac{1}{2}\alpha+\frac{1}{4}}\frac{1-q^{n+1}}{1-q} {\tilde w}(x;q^{\alpha-1}|q)\ctsqLaguerre{\alpha-1}{n+1}@{x}{q} }

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

w ⁢ ( x ) := w ⁢ ( x ; q α | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ ⁢ q 1 2 ⁢ α + 3 4 ⁢ e i ⁢ θ ; q ) ∞ | 2 = | ( e i ⁢ θ , - e i ⁢ θ ; q 1 2 ) ∞ ( q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ ; q 1 2 ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , q 1 2 ⁢ α + 1 4 ) ⁢ h ⁢ ( x , q 1 2 ⁢ α + 3 4 ) assign 𝑤 𝑥 𝑤 𝑥 conditional superscript 𝑞 𝛼 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛼 3 4 imaginary-unit 𝜃 𝑞 2 superscript q-Pochhammer-symbol imaginary-unit 𝜃 imaginary-unit 𝜃 superscript 𝑞 1 2 q-Pochhammer-symbol superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 𝛼 1 4 ℎ 𝑥 superscript 𝑞 1 2 𝛼 3 4 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha}|q)=\left|\frac% {\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(q^{\frac{1}{2% }\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}q^{\frac{1}{2}\alpha+\frac{% 3}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\left|% \frac{\left({\mathrm{e}^{\mathrm{i}\theta}},-{\mathrm{e}^{\mathrm{i}\theta}};q% ^{\frac{1}{2}}\right)_{\infty}}{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}{% \mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\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,q^{\frac{1}{% 2}\alpha+\frac{1}{4}})h(x,q^{\frac{1}{2}\alpha+\frac{3}{4}})}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; 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 𝛼 imaginary-unit 𝜃 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}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


Rodrigues-type formula

w ~ ⁢ ( x ; q α | q ) ⁢ P n ( α ) ⁡ ( x | q ) = ( q - 1 2 ) n ⁢ q 1 4 ⁢ n 2 + 1 2 ⁢ n ⁢ α ( q ; q ) n ⁢ ( D q ) n ⁢ [ w ~ ⁢ ( x ; q α + n | q ) ] ~ 𝑤 𝑥 conditional superscript 𝑞 𝛼 𝑞 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 superscript 𝑞 1 2 𝑛 superscript 𝑞 1 4 superscript 𝑛 2 1 2 𝑛 𝛼 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript subscript 𝐷 𝑞 𝑛 delimited-[] ~ 𝑤 𝑥 conditional superscript 𝑞 𝛼 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;q^{\alpha}|q)P^{(% \alpha)}_{n}\!\left(x|q\right)=\left(\frac{q-1}{2}\right)^{n}\frac{q^{\frac{1}% {4}n^{2}+\frac{1}{2}n\alpha}}{\left(q;q\right)_{n}}\left(D_{q}\right)^{n}\left% [{\tilde{w}}(x;q^{\alpha+n}|q)\right]}}} {\displaystyle {\tilde w}(x;q^{\alpha}|q)\ctsqLaguerre{\alpha}{n}@{x}{q}=\left(\frac{q-1}{2}\right)^n \frac{q^{\frac{1}{4}n^2+\frac{1}{2}n\alpha}}{\qPochhammer{q}{q}{n}} \left(D_q\right)^n\left[{\tilde w}(x;q^{\alpha+n}|q)\right] }

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

w ⁢ ( x ) := w ⁢ ( x ; q α | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ ⁢ q 1 2 ⁢ α + 3 4 ⁢ e i ⁢ θ ; q ) ∞ | 2 = | ( e i ⁢ θ , - e i ⁢ θ ; q 1 2 ) ∞ ( q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ ; q 1 2 ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , q 1 2 ⁢ α + 1 4 ) ⁢ h ⁢ ( x , q 1 2 ⁢ α + 3 4 ) assign 𝑤 𝑥 𝑤 𝑥 conditional superscript 𝑞 𝛼 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛼 3 4 imaginary-unit 𝜃 𝑞 2 superscript q-Pochhammer-symbol imaginary-unit 𝜃 imaginary-unit 𝜃 superscript 𝑞 1 2 q-Pochhammer-symbol superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 𝛼 1 4 ℎ 𝑥 superscript 𝑞 1 2 𝛼 3 4 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha}|q)=\left|\frac% {\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(q^{\frac{1}{2% }\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}q^{\frac{1}{2}\alpha+\frac{% 3}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\left|% \frac{\left({\mathrm{e}^{\mathrm{i}\theta}},-{\mathrm{e}^{\mathrm{i}\theta}};q% ^{\frac{1}{2}}\right)_{\infty}}{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}{% \mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\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,q^{\frac{1}{% 2}\alpha+\frac{1}{4}})h(x,q^{\frac{1}{2}\alpha+\frac{3}{4}})}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; 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 𝛼 imaginary-unit 𝜃 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}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


Generating functions

( q α + 1 2 ⁢ t , q α + 1 ⁢ t ; q ) ∞ ( q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ ⁢ t , q 1 2 ⁢ α + 1 4 ⁢ e - i ⁢ θ ⁢ t ; q ) ∞ = ∑ n = 0 ∞ P n ( α ) ⁡ ( x | q ) ⁢ t n q-Pochhammer-symbol superscript 𝑞 𝛼 1 2 𝑡 superscript 𝑞 𝛼 1 𝑡 𝑞 q-Pochhammer-symbol superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 𝑡 superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 𝑡 𝑞 superscript subscript 𝑛 0 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{\left(q^{\alpha+\frac{1}{2}}t,% q^{\alpha+1}t;q\right)_{\infty}}{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}{% \mathrm{e}^{\mathrm{i}\theta}}t,q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{% -\mathrm{i}\theta}}t;q\right)_{\infty}}=\sum_{n=0}^{\infty}P^{(\alpha)}_{n}\!% \left(x|q\right)t^{n}}}} {\displaystyle \frac{\qPochhammer{q^{\alpha+\frac{1}{2}}t,q^{\alpha+1}t}{q}{\infty}} {\qPochhammer{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta}t,q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{-\iunit\theta}t}{q}{\infty}} =\sum_{n=0}^{\infty}\ctsqLaguerre{\alpha}{n}@{x}{q}t^n }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


1 ( e i ⁢ θ ⁢ t ; q ) ∞ ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ ⁢ q 1 2 ⁢ α + 3 4 ⁢ e i ⁢ θ ⁢ q α + 1 ⁢ q ⁢ e - i ⁢ θ ⁢ t = ∑ n = 0 ∞ P n ( α ) ⁡ ( x | q ) ⁢ t n ( q α + 1 ; q ) n ⁢ q ( 1 2 ⁢ α + 1 4 ) ⁢ n 1 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 𝑞 \qHyperrphis 21 @ @ superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛼 3 4 imaginary-unit 𝜃 superscript 𝑞 𝛼 1 𝑞 imaginary-unit 𝜃 𝑡 superscript subscript 𝑛 0 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 superscript 𝑡 𝑛 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 superscript 𝑞 1 2 𝛼 1 4 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{\left({\mathrm{e}^{\mathrm{% i}\theta}}t;q\right)_{\infty}}\,\qHyperrphis{2}{1}@@{q^{\frac{1}{2}\alpha+% \frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}q^{\frac{1}{2}\alpha+\frac{3}{4}}{% \mathrm{e}^{\mathrm{i}\theta}}}{q^{\alpha+1}}{q}{{\mathrm{e}^{-\mathrm{i}% \theta}}t}{}=\sum_{n=0}^{\infty}\frac{P^{(\alpha)}_{n}\!\left(x|q\right)t^{n}}% {\left(q^{\alpha+1};q\right)_{n}q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}}}} {\displaystyle \frac{1}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\,\qHyperrphis{2}{1}@@{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta} q^{\frac{1}{2}\alpha+\frac{3}{4}}\expe^{\iunit\theta}}{q^{\alpha+1}}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\ctsqLaguerre{\alpha}{n}@{x}{q}t^n}{\qPochhammer{q^{\alpha+1}}{q}{n}q^{(\frac{1}{2}\alpha+\frac{1}{4})n}} }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


( t ; q ) ∞ ⋅ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ ⁢ q 1 2 ⁢ α + 1 4 ⁢ e - i ⁢ θ ⁢ q α + 1 ⁢ q ⁢ t = ∑ n = 0 ∞ ( - 1 ) n ⁢ q \binomial ⁢ n ⁢ 2 ( q α + 1 ; q ) n ⁢ P n ( α ) ⁡ ( x | q ) ⁢ t n ⋅ q-Pochhammer-symbol 𝑡 𝑞 \qHyperrphis 21 @ @ superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 𝛼 1 𝑞 𝑡 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left(t;q\right)_{\infty}\cdot% \qHyperrphis{2}{1}@@{q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}% \theta}}q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{-\mathrm{i}\theta}}}{q^{% \alpha+1}}{q}{t}{}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binomial{n}{2}}}{\left% (q^{\alpha+1};q\right)_{n}}P^{(\alpha)}_{n}\!\left(x|q\right)t^{n}}}} {\displaystyle \qPochhammer{t}{q}{\infty}\cdot\qHyperrphis{2}{1}@@{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta} q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{-\iunit\theta}}{q^{\alpha+1}}{q}{t} {}=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{n}{2}}}{\qPochhammer{q^{\alpha+1}}{q}{n}} \ctsqLaguerre{\alpha}{n}@{x}{q}t^n }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


( γ ⁢ e i ⁢ θ ⁢ t ; q ) ∞ ( e i ⁢ θ ⁢ t ; q ) ∞ ⁢ \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ γ , q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ , q 1 2 ⁢ α + 3 4 ⁢ e i ⁢ θ ⁢ q α + 1 , γ ⁢ e i ⁢ θ ⁢ t ⁢ q ⁢ e - i ⁢ θ ⁢ t = ∑ n = 0 ∞ ( γ ; q ) n ( q α + 1 ; q ) n ⁢ P n ⁡ ( x | q ) q ( 1 2 ⁢ α + 1 4 ) ⁢ n ⁢ t n q-Pochhammer-symbol 𝛾 imaginary-unit 𝜃 𝑡 𝑞 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 𝑞 \qHyperrphis 32 @ @ 𝛾 superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛼 3 4 imaginary-unit 𝜃 superscript 𝑞 𝛼 1 𝛾 imaginary-unit 𝜃 𝑡 𝑞 imaginary-unit 𝜃 𝑡 superscript subscript 𝑛 0 q-Pochhammer-symbol 𝛾 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 continuous-q-Legendre-polynomial-P 𝑛 𝑥 𝑞 superscript 𝑞 1 2 𝛼 1 4 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{\left(\gamma{\mathrm{e}^{% \mathrm{i}\theta}}t;q\right)_{\infty}}{\left({\mathrm{e}^{\mathrm{i}\theta}}t;% q\right)_{\infty}}\ \qHyperrphis{3}{2}@@{\gamma,q^{\frac{1}{2}\alpha+\frac{1}{% 4}}{\mathrm{e}^{\mathrm{i}\theta}},q^{\frac{1}{2}\alpha+\frac{3}{4}}{\mathrm{e% }^{\mathrm{i}\theta}}}{q^{\alpha+1},\gamma{\mathrm{e}^{\mathrm{i}\theta}}t}{q}% {{\mathrm{e}^{-\mathrm{i}\theta}}t}{}=\sum_{n=0}^{\infty}\frac{\left(\gamma;q% \right)_{n}}{\left(q^{\alpha+1};q\right)_{n}}\frac{P_{n}\!\left(x|q\right)}{q^% {(\frac{1}{2}\alpha+\frac{1}{4})n}}t^{n}}}} {\displaystyle \frac{\qPochhammer{\gamma\expe^{\iunit\theta}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ \qHyperrphis{3}{2}@@{\gamma,q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta},q^{\frac{1}{2}\alpha+\frac{3}{4}}\expe^{\iunit\theta}}{q^{\alpha+1},\gamma\expe^{\iunit\theta}t}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{\gamma}{q}{n}}{\qPochhammer{q^{\alpha+1}}{q}{n}}\frac{\ctsqLegendre{n}@{x}{q}}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}t^n }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}} &
γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


Limit relations

Al-Salam-Chihara polynomial to Continuous q-Laguerre polynomial

Q n ⁡ ( x ; q 1 2 ⁢ α + 1 4 , q 1 2 ⁢ α + 3 4 | q ) = ( q ; q ) n q ( 1 2 ⁢ α + 1 4 ) ⁢ n ⁢ P n ( α ) ⁡ ( x | q ) Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 superscript 𝑞 1 2 𝛼 1 4 superscript 𝑞 1 2 𝛼 3 4 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 1 2 𝛼 1 4 𝑛 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;q^{\frac{1}{2}\alpha% +\frac{1}{4}},q^{\frac{1}{2}\alpha+\frac{3}{4}}\,|\,q\right)=\frac{\left(q;q% \right)_{n}}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}P^{(\alpha)}_{n}\!\left(x|q% \right)}}} {\displaystyle \AlSalamChihara{n}@{x}{q^{\frac{1}{2}\alpha+\frac{1}{4}}}{q^{\frac{1}{2}\alpha+\frac{3}{4}}}{q} =\frac{\qPochhammer{q}{q}{n}}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}\ctsqLaguerre{\alpha}{n}@{x}{q} }

q-Meixner-Pollaczek polynomial to Continuous q-Laguerre polynomial

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 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} }

Continuous q-Jacobi polynomial to Continuous q-Laguerre polynomial

lim β → ∞ ⁡ P n ( α , β ) ⁡ ( x | q ) = P n ( α ) ⁡ ( x | q ) subscript → 𝛽 continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{\beta\rightarrow\infty}P^{(% \alpha,\beta)}_{n}\!\left(x|q\right)=P^{(\alpha)}_{n}\!\left(x|q\right)}}} {\displaystyle \lim _{\beta\rightarrow\infty} \ctsqJacobi{\alpha}{\beta}{n}@{x}{q}=\ctsqLaguerre{\alpha}{n}@{x}{q} }

Continuous q-Laguerre polynomial to Continuous q-Hermite polynomial

lim α → ∞ ⁡ P n ( α ) ⁡ ( x | q ) q ( 1 2 ⁢ α + 1 4 ) ⁢ n = H n ⁡ ( x | q ) ( q ; q ) n subscript → 𝛼 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 superscript 𝑞 1 2 𝛼 1 4 𝑛 continuous-q-Hermite-polynomial-H 𝑛 𝑥 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}\frac{% P^{(\alpha)}_{n}\!\left(x|q\right)}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}=% \frac{H_{n}\!\left(x\,|\,q\right)}{\left(q;q\right)_{n}}}}} {\displaystyle \lim_{\alpha\rightarrow\infty} \frac{\ctsqLaguerre{\alpha}{n}@{x}{q}}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}} =\frac{\ctsqHermite{n}@{x}{q}}{\qPochhammer{q}{q}{n}} }

Continuous q-Laguerre polynomial to Laguerre polynomial

lim q → 1 ⁡ P n ( α ) ⁡ ( q x | q ) = L n α ⁡ ( 2 ⁢ x ) subscript → 𝑞 1 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 superscript 𝑞 𝑥 𝑞 generalized-Laguerre-polynomial-L 𝛼 𝑛 2 𝑥 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}P^{(\alpha)}_{n% }\!\left(q^{x}|q\right)=L^{\alpha}_{n}\left(2x\right)}}} {\displaystyle \lim_{q\rightarrow 1}\ctsqLaguerre{\alpha}{n}@{q^x}{q}=\Laguerre[\alpha]{n}@{2x} }

Remark

P n ( α ) ⁡ ( x ; q ) = ( q α + 1 ; q ) n ( q ; q ) n ⁢ \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ q - n , q 1 2 ⁢ e i ⁢ θ , q 1 2 ⁢ e - i ⁢ θ ⁢ q α + 1 , - q ⁢ q ⁢ q Meixner-Pollaczek-polynomial-P 𝛼 𝑛 𝑥 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 superscript 𝑞 1 2 imaginary-unit 𝜃 superscript 𝑞 1 2 imaginary-unit 𝜃 superscript 𝑞 𝛼 1 𝑞 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha)}_{n}\!\left(x;q\right)=% \frac{\left(q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}}\,\qHyperrphis{3}{% 2}@@{q^{-n},q^{\frac{1}{2}}{\mathrm{e}^{\mathrm{i}\theta}},q^{\frac{1}{2}}{% \mathrm{e}^{-\mathrm{i}\theta}}}{q^{\alpha+1},-q}{q}{q}}}} {\displaystyle \MeixnerPollaczek{\alpha}{n}@{x}{q}=\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}}\,\qHyperrphis{3}{2}@@{q^{-n},q^{\frac{1}{2}}\expe^{\iunit\theta},q^{\frac{1}{2}}\expe^{-\iunit\theta}}{q^{\alpha+1},-q}{q}{q} }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


P n ( α ) ⁡ ( x | q 2 ) = q n ⁢ α ⁢ P n ( α ) ⁡ ( x ; q ) continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 superscript 𝑞 2 superscript 𝑞 𝑛 𝛼 Meixner-Pollaczek-polynomial-P 𝛼 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha)}_{n}\!\left(x|q^{2}% \right)=q^{n\alpha}P^{(\alpha)}_{n}\!\left(x;q\right)}}} {\displaystyle \ctsqLaguerre{\alpha}{n}@{x}{q^2}=q^{n\alpha}\MeixnerPollaczek{\alpha}{n}@{x}{q} }

Reference

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