Continuous q-Jacobi: Difference between revisions

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Latest revision as of 00:33, 6 March 2017

Continuous q-Jacobi

Basic hypergeometric representation

P n ( α , β ) ⁡ ( x | q ) = ( q α + 1 ; q ) n ( q ; q ) n ⁢ \qHyperrphis ⁢ 43 ⁢ @ ⁢ @ ⁢ q - n , q n + α + β + 1 , q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ , q 1 2 ⁢ α + 1 4 ⁢ e - i ⁢ θ ⁢ q α + 1 , - q 1 2 ⁢ ( α + β + 1 ) , - q 1 2 ⁢ ( α + β + 2 ) ⁢ q ⁢ q continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 \qHyperrphis 43 @ @ superscript 𝑞 𝑛 superscript 𝑞 𝑛 𝛼 𝛽 1 superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 𝛼 1 superscript 𝑞 1 2 𝛼 𝛽 1 superscript 𝑞 1 2 𝛼 𝛽 2 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\!\left(x|q% \right){}=\frac{\left(q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}}\ % \qHyperrphis{4}{3}@@{q^{-n},q^{n+\alpha+\beta+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^{\frac{1}{2}(\alpha+\beta+1)},-q^{% \frac{1}{2}(\alpha+\beta+2)}}{q}{q}}}} {\displaystyle \ctsqJacobi{\alpha}{\beta}{n}@{x}{q} {}=\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}}\ \qHyperrphis{4}{3}@@{q^{-n},q^{n+\alpha+\beta+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^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{q} }

Orthogonality relation(s)

1 2 ⁢ π ⁢ ∫ - 1 1 w ⁢ ( x ) 1 - x 2 ⁢ P m ( α , β ) ⁡ ( x | q ) ⁢ P n ( α , β ) ⁡ ( x | q ) ⁢ 𝑑 x = ( q 1 2 ⁢ ( α + β + 2 ) , q 1 2 ⁢ ( α + β + 3 ) ; q ) ∞ ( q , q α + 1 , q β + 1 , - q 1 2 ⁢ ( α + β + 1 ) - q 1 2 ⁢ ( α + β + 2 ) ; q ) ∞ ⁢ 1 - q α + β + 1 1 - q 2 ⁢ n + α + β + 1 ⁢ ( q α + 1 , q β + 1 , - q 1 2 ⁢ ( α + β + 3 ) ; q ) n ( q , q α + β + 1 , - q 1 2 ⁢ ( α + β + 1 ) ; q ) n ⁢ q ( α + 1 2 ) ⁢ n ⁢ δ m , n 1 2 superscript subscript 1 1 𝑤 𝑥 1 superscript 𝑥 2 continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑚 𝑥 𝑞 continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 differential-d 𝑥 q-Pochhammer-symbol superscript 𝑞 1 2 𝛼 𝛽 2 superscript 𝑞 1 2 𝛼 𝛽 3 𝑞 q-Pochhammer-symbol 𝑞 superscript 𝑞 𝛼 1 superscript 𝑞 𝛽 1 superscript 𝑞 1 2 𝛼 𝛽 1 superscript 𝑞 1 2 𝛼 𝛽 2 𝑞 1 superscript 𝑞 𝛼 𝛽 1 1 superscript 𝑞 2 𝑛 𝛼 𝛽 1 q-Pochhammer-symbol superscript 𝑞 𝛼 1 superscript 𝑞 𝛽 1 superscript 𝑞 1 2 𝛼 𝛽 3 𝑞 𝑛 q-Pochhammer-symbol 𝑞 superscript 𝑞 𝛼 𝛽 1 superscript 𝑞 1 2 𝛼 𝛽 1 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,\beta)}_{m}\!\left(x|q\right)P^{(\alpha,\beta)}_{% n}\!\left(x|q\right)\,dx{}=\frac{\left(q^{\frac{1}{2}(\alpha+\beta+2)},q^{% \frac{1}{2}(\alpha+\beta+3)};q\right)_{\infty}}{\left(q,q^{\alpha+1},q^{\beta+% 1},-q^{\frac{1}{2}(\alpha+\beta+1)}-q^{\frac{1}{2}(\alpha+\beta+2)};q\right)_{% \infty}}\,\frac{1-q^{\alpha+\beta+1}}{1-q^{2n+\alpha+\beta+1}}{}\frac{\left(q^% {\alpha+1},q^{\beta+1},-q^{\frac{1}{2}(\alpha+\beta+3)};q\right)_{n}}{\left(q,% q^{\alpha+\beta+1},-q^{\frac{1}{2}(\alpha+\beta+1)};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}} \ctsqJacobi{\alpha}{\beta}{m}@{x}{q}\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}\,dx {}=\frac{\qPochhammer{q^{\frac{1}{2}(\alpha+\beta+2)},q^{\frac{1}{2}(\alpha+\beta+3)}}{q}{\infty}}{\qPochhammer{q,q^{\alpha+1},q^{\beta+1},-q^{\frac{1}{2}(\alpha+\beta+1)} -q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{\infty}}\,\frac{1-q^{\alpha+\beta+1}}{1-q^{2n+\alpha+\beta+1}} {}\frac{\qPochhammer{q^{\alpha+1},q^{\beta+1},-q^{\frac{1}{2}(\alpha+\beta+3)}}{q}{n}} {\qPochhammer{q,q^{\alpha+\beta+1},-q^{\frac{1}{2}(\alpha+\beta+1)}}{q}{n}}q^{(\alpha+\frac{1}{2})n}\,\Kronecker{m}{n} }

Substitution(s): w ⁢ ( x ) := w ⁢ ( x ; q α , q β | 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 1 2 ⁢ β + 1 4 ⁢ e i ⁢ θ , - q 1 2 ⁢ β + 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 ⁢ β + 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 ) ⁢ h ⁢ ( x , - q 1 2 ⁢ β + 1 4 ) ⁢ h ⁢ ( x , - q 1 2 ⁢ β + 3 4 ) assign 𝑤 𝑥 𝑤 𝑥 superscript 𝑞 𝛼 conditional superscript 𝑞 𝛽 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 superscript 𝑞 1 2 𝛽 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛽 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 𝛽 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 ℎ 𝑥 superscript 𝑞 1 2 𝛽 1 4 ℎ 𝑥 superscript 𝑞 1 2 𝛽 3 4 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha},q^{\beta}|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^{\frac{1}{2}\beta+\frac{1% }{4}}{\mathrm{e}^{\mathrm{i}\theta}},-q^{\frac{1}{2}\beta+\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}\beta+\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,-q^{\frac{1}{2}\beta+\frac{1}{4}})h(x,-q^{\frac{1}% {2}\beta+\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 ( n , β ) ⁡ ( x | q ) = A n ⁢ P ~ n ( n + 1 , β ) ⁡ ( x | q ) + [ q 1 2 ⁢ α + 1 4 + q - 1 2 ⁢ α - 1 4 - ( A n + C n ) ] ⁢ P ~ n ( n , β ) ⁡ ( x | q ) + C n ⁢ P ~ n ( n - 1 , β ) ⁡ ( x | q ) 2 𝑥 continuous-q-Jacobi-polynomial-P-tilde 𝑛 𝛽 𝑛 𝑥 𝑞 subscript 𝐴 𝑛 continuous-q-Jacobi-polynomial-P-tilde 𝑛 1 𝛽 𝑛 𝑥 𝑞 delimited-[] superscript 𝑞 1 2 𝛼 1 4 superscript 𝑞 1 2 𝛼 1 4 subscript 𝐴 𝑛 subscript 𝐶 𝑛 continuous-q-Jacobi-polynomial-P-tilde 𝑛 𝛽 𝑛 𝑥 𝑞 subscript 𝐶 𝑛 continuous-q-Jacobi-polynomial-P-tilde 𝑛 1 𝛽 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle 2x{\tilde{P}}^{(n,\beta)}_{n}\!% \left(x|q\right)=A_{n}{\tilde{P}}^{(n+1,\beta)}_{n}\!\left(x|q\right)+\left[q^% {\frac{1}{2}\alpha+\frac{1}{4}}+q^{-\frac{1}{2}\alpha-\frac{1}{4}}-\left(A_{n}% +C_{n}\right)\right]{\tilde{P}}^{(n,\beta)}_{n}\!\left(x|q\right){}+C_{n}{% \tilde{P}}^{(n-1,\beta)}_{n}\!\left(x|q\right)}}} {\displaystyle 2x\normctsqJacobiPtilde{n}{\beta}{n}@@{x}{q}=A_n\normctsqJacobiPtilde{n+1}{\beta}{n}@@{x}{q}+\left[q^{\frac{1}{2}\alpha+\frac{1}{4}}+ q^{-\frac{1}{2}\alpha-\frac{1}{4}}-\left(A_n+C_n\right)\right]\normctsqJacobiPtilde{n}{\beta}{n}@@{x}{q} {}+C_n\normctsqJacobiPtilde{n-1}{\beta}{n}@@{x}{q} }

Substitution(s): C n = q 1 2 ⁢ α + 1 4 ⁢ ( 1 - q n ) ⁢ ( 1 - q n + β ) ⁢ ( 1 + q n + 1 2 ⁢ ( α + β ) ) ⁢ ( 1 + q n + 1 2 ⁢ ( α + β + 1 ) ) ( 1 - q 2 ⁢ n + α + β ) ⁢ ( 1 - q 2 ⁢ n + α + β + 1 ) subscript 𝐶 𝑛 superscript 𝑞 1 2 𝛼 1 4 1 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 𝛽 1 superscript 𝑞 𝑛 1 2 𝛼 𝛽 1 superscript 𝑞 𝑛 1 2 𝛼 𝛽 1 1 superscript 𝑞 2 𝑛 𝛼 𝛽 1 superscript 𝑞 2 𝑛 𝛼 𝛽 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{q^{\frac{1}{2}\alpha+% \frac{1}{4}}(1-q^{n})(1-q^{n+\beta})(1+q^{n+\frac{1}{2}(\alpha+\beta)})(1+q^{n% +\frac{1}{2}(\alpha+\beta+1)})}{(1-q^{2n+\alpha+\beta})(1-q^{2n+\alpha+\beta+1% })}}}} &
A n = ( 1 - q n + α + 1 ) ⁢ ( 1 - q n + α + β + 1 ) ⁢ ( 1 + q n + 1 2 ⁢ ( α + β + 1 ) ) ⁢ ( 1 + q n + 1 2 ⁢ ( α + β + 2 ) ) q 1 2 ⁢ α + 1 4 ⁢ ( 1 - q 2 ⁢ n + α + β + 1 ) ⁢ ( 1 - q 2 ⁢ n + α + β + 2 ) subscript 𝐴 𝑛 1 superscript 𝑞 𝑛 𝛼 1 1 superscript 𝑞 𝑛 𝛼 𝛽 1 1 superscript 𝑞 𝑛 1 2 𝛼 𝛽 1 1 superscript 𝑞 𝑛 1 2 𝛼 𝛽 2 superscript 𝑞 1 2 𝛼 1 4 1 superscript 𝑞 2 𝑛 𝛼 𝛽 1 1 superscript 𝑞 2 𝑛 𝛼 𝛽 2 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n+\alpha+1})(1-q^% {n+\alpha+\beta+1})(1+q^{n+\frac{1}{2}(\alpha+\beta+1)})(1+q^{n+\frac{1}{2}(% \alpha+\beta+2)})}{q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^{2n+\alpha+\beta+1})(% 1-q^{2n+\alpha+\beta+2})}}}}


P ~ n ( n , β ) ⁡ ( x | q ) := P ~ n ( α , β ) ⁡ ( x | q ) = ( q ; q ) n ( q α + 1 ; q ) n ⁢ P n ( α , β ) ⁡ ( x | q ) assign continuous-q-Jacobi-polynomial-P-tilde 𝑛 𝛽 𝑛 𝑥 𝑞 continuous-q-Jacobi-polynomial-P-tilde 𝛼 𝛽 𝑛 𝑥 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle{\tilde{P}}^{(n,\beta)}_{n}\!\left(x% |q\right):={\tilde{P}}^{(\alpha,\beta)}_{n}\!\left(x|q\right)=\frac{\left(q;q% \right)_{n}}{\left(q^{\alpha+1};q\right)_{n}}P^{(\alpha,\beta)}_{n}\!\left(x|q% \right)}}} {\displaystyle \normctsqJacobiPtilde{n}{\beta}{n}@@{x}{q}:=\normctsqJacobiPtilde{\alpha}{\beta}{n}@{x}{q}=\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q^{\alpha+1}}{q}{n}}\ctsqJacobi{\alpha}{\beta}{n}@{x}{q} }

Monic recurrence relation

x ⁢ P ^ n ( α , β ) ⁡ ( x ) = P ^ n + 1 ( α , β ) ⁡ ( x ) + 1 2 ⁢ [ q 1 2 ⁢ α + 1 4 + q - 1 2 ⁢ α - 1 4 - ( A n + C n ) ] ⁢ P ^ n ( α , β ) ⁡ ( x ) + 1 4 ⁢ A n - 1 ⁢ C n ⁢ P ^ n - 1 ( α , β ) ⁡ ( x ) 𝑥 continuous-q-Jacobi-polynomial-monic-p 𝛼 𝛽 𝑛 𝑥 𝑞 continuous-q-Jacobi-polynomial-monic-p 𝛼 𝛽 𝑛 1 𝑥 𝑞 1 2 delimited-[] superscript 𝑞 1 2 𝛼 1 4 superscript 𝑞 1 2 𝛼 1 4 subscript 𝐴 𝑛 subscript 𝐶 𝑛 continuous-q-Jacobi-polynomial-monic-p 𝛼 𝛽 𝑛 𝑥 𝑞 1 4 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 continuous-q-Jacobi-polynomial-monic-p 𝛼 𝛽 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}^{(\alpha,\beta)}_{n}% \!\left(x\right)={\widehat{P}}^{(\alpha,\beta)}_{n+1}\!\left(x\right)+\frac{1}% {2}\left[q^{\frac{1}{2}\alpha+\frac{1}{4}}+q^{-\frac{1}{2}\alpha-\frac{1}{4}}-% (A_{n}+C_{n})\right]{\widehat{P}}^{(\alpha,\beta)}_{n}\!\left(x\right){}+\frac% {1}{4}A_{n-1}C_{n}{\widehat{P}}^{(\alpha,\beta)}_{n-1}\!\left(x\right)}}} {\displaystyle x\monicctsqJacobi{\alpha}{\beta}{n}@@{x}{q}=\monicctsqJacobi{\alpha}{\beta}{n+1}@@{x}{q}+\frac{1}{2}\left[q^{\frac{1}{2}\alpha+\frac{1}{4}}+ q^{-\frac{1}{2}\alpha-\frac{1}{4}}-(A_n+C_n)\right]\monicctsqJacobi{\alpha}{\beta}{n}@@{x}{q} {}+\frac{1}{4}A_{n-1}C_n\monicctsqJacobi{\alpha}{\beta}{n-1}@@{x}{q} }

Substitution(s): C n = q 1 2 ⁢ α + 1 4 ⁢ ( 1 - q n ) ⁢ ( 1 - q n + β ) ⁢ ( 1 + q n + 1 2 ⁢ ( α + β ) ) ⁢ ( 1 + q n + 1 2 ⁢ ( α + β + 1 ) ) ( 1 - q 2 ⁢ n + α + β ) ⁢ ( 1 - q 2 ⁢ n + α + β + 1 ) subscript 𝐶 𝑛 superscript 𝑞 1 2 𝛼 1 4 1 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 𝛽 1 superscript 𝑞 𝑛 1 2 𝛼 𝛽 1 superscript 𝑞 𝑛 1 2 𝛼 𝛽 1 1 superscript 𝑞 2 𝑛 𝛼 𝛽 1 superscript 𝑞 2 𝑛 𝛼 𝛽 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{q^{\frac{1}{2}\alpha+% \frac{1}{4}}(1-q^{n})(1-q^{n+\beta})(1+q^{n+\frac{1}{2}(\alpha+\beta)})(1+q^{n% +\frac{1}{2}(\alpha+\beta+1)})}{(1-q^{2n+\alpha+\beta})(1-q^{2n+\alpha+\beta+1% })}}}} &
A n = ( 1 - q n + α + 1 ) ⁢ ( 1 - q n + α + β + 1 ) ⁢ ( 1 + q n + 1 2 ⁢ ( α + β + 1 ) ) ⁢ ( 1 + q n + 1 2 ⁢ ( α + β + 2 ) ) q 1 2 ⁢ α + 1 4 ⁢ ( 1 - q 2 ⁢ n + α + β + 1 ) ⁢ ( 1 - q 2 ⁢ n + α + β + 2 ) subscript 𝐴 𝑛 1 superscript 𝑞 𝑛 𝛼 1 1 superscript 𝑞 𝑛 𝛼 𝛽 1 1 superscript 𝑞 𝑛 1 2 𝛼 𝛽 1 1 superscript 𝑞 𝑛 1 2 𝛼 𝛽 2 superscript 𝑞 1 2 𝛼 1 4 1 superscript 𝑞 2 𝑛 𝛼 𝛽 1 1 superscript 𝑞 2 𝑛 𝛼 𝛽 2 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n+\alpha+1})(1-q^% {n+\alpha+\beta+1})(1+q^{n+\frac{1}{2}(\alpha+\beta+1)})(1+q^{n+\frac{1}{2}(% \alpha+\beta+2)})}{q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^{2n+\alpha+\beta+1})(% 1-q^{2n+\alpha+\beta+2})}}}}


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

q-Difference equation

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

Substitution(s): λ n = 4 ⁢ q - n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 - q n + α + β + 1 ) subscript 𝜆 𝑛 4 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 𝛼 𝛽 1 {\displaystyle{\displaystyle{\displaystyle\lambda_{n}=4q^{-n+1}(1-q^{n})(1-q^{% n+\alpha+\beta+1})}}} &

w ~ ⁢ ( x ; q α , q β | q ) := w ⁢ ( x ; q α , q β | q ) 1 - x 2 assign ~ 𝑤 𝑥 superscript 𝑞 𝛼 conditional superscript 𝑞 𝛽 𝑞 𝑤 𝑥 superscript 𝑞 𝛼 conditional superscript 𝑞 𝛽 𝑞 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;q^{\alpha},q^{\beta}|q% ):=\frac{w(x;q^{\alpha},q^{\beta}|q)}{\sqrt{1-x^{2}}}}}} &
y ⁢ ( x ) = P n ( α , β ) ⁡ ( x | q ) 𝑦 𝑥 continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=P^{(\alpha,\beta)}_{n}\!\left(% x|q\right)}}} &
w ⁢ ( x ) := w ⁢ ( x ; q α , q β | 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 1 2 ⁢ β + 1 4 ⁢ e i ⁢ θ , - q 1 2 ⁢ β + 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 ⁢ β + 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 ) ⁢ h ⁢ ( x , - q 1 2 ⁢ β + 1 4 ) ⁢ h ⁢ ( x , - q 1 2 ⁢ β + 3 4 ) assign 𝑤 𝑥 𝑤 𝑥 superscript 𝑞 𝛼 conditional superscript 𝑞 𝛽 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 superscript 𝑞 1 2 𝛽 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛽 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 𝛽 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 ℎ 𝑥 superscript 𝑞 1 2 𝛽 1 4 ℎ 𝑥 superscript 𝑞 1 2 𝛽 3 4 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha},q^{\beta}|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^{\frac{1}{2}\beta+\frac{1% }{4}}{\mathrm{e}^{\mathrm{i}\theta}},-q^{\frac{1}{2}\beta+\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}\beta+\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,-q^{\frac{1}{2}\beta+\frac{1}{4}})h(x,-q^{\frac{1}% {2}\beta+\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 , β ) ⁡ ( x | q ) = - q - n + 1 2 ⁢ α + 3 4 ⁢ ( 1 - q n + α + β + 1 ) ⁢ ( e i ⁢ θ - e - i ⁢ θ ) ( 1 + q 1 2 ⁢ ( α + β + 1 ) ) ⁢ ( 1 + q 1 2 ⁢ ( α + β + 2 ) ) ⁢ P n - 1 ( α + 1 , β + 1 ) ⁢ ( x | q ) subscript 𝛿 𝑞 continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 superscript 𝑞 𝑛 1 2 𝛼 3 4 1 superscript 𝑞 𝑛 𝛼 𝛽 1 imaginary-unit 𝜃 imaginary-unit 𝜃 1 superscript 𝑞 1 2 𝛼 𝛽 1 1 superscript 𝑞 1 2 𝛼 𝛽 2 superscript subscript 𝑃 𝑛 1 𝛼 1 𝛽 1 conditional 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}P^{(\alpha,\beta)}_{n}\!% \left(x|q\right)=-\frac{q^{-n+\frac{1}{2}\alpha+\frac{3}{4}}(1-q^{n+\alpha+% \beta+1})({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e}^{-\mathrm{i}\theta}})}{(% 1+q^{\frac{1}{2}(\alpha+\beta+1)})(1+q^{\frac{1}{2}(\alpha+\beta+2)})}{}P_{n-1% }^{(\alpha+1,\beta+1)}(x|q)}}} {\displaystyle \delta_q\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}=-\frac{q^{-n+\frac{1}{2}\alpha+\frac{3}{4}} (1-q^{n+\alpha+\beta+1})(\expe^{\iunit\theta}-\expe^{-\iunit\theta})}{(1+q^{\frac{1}{2}(\alpha+\beta+1)}) (1+q^{\frac{1}{2}(\alpha+\beta+2)})} {} P_{n-1}^{(\alpha+1,\beta+1)}(x|q) }

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


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

Backward shift operator

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

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

w ⁢ ( x ) := w ⁢ ( x ; q α , q β | 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 1 2 ⁢ β + 1 4 ⁢ e i ⁢ θ , - q 1 2 ⁢ β + 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 ⁢ β + 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 ) ⁢ h ⁢ ( x , - q 1 2 ⁢ β + 1 4 ) ⁢ h ⁢ ( x , - q 1 2 ⁢ β + 3 4 ) assign 𝑤 𝑥 𝑤 𝑥 superscript 𝑞 𝛼 conditional superscript 𝑞 𝛽 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 superscript 𝑞 1 2 𝛽 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛽 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 𝛽 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 ℎ 𝑥 superscript 𝑞 1 2 𝛽 1 4 ℎ 𝑥 superscript 𝑞 1 2 𝛽 3 4 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha},q^{\beta}|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^{\frac{1}{2}\beta+\frac{1% }{4}}{\mathrm{e}^{\mathrm{i}\theta}},-q^{\frac{1}{2}\beta+\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}\beta+\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,-q^{\frac{1}{2}\beta+\frac{1}{4}})h(x,-q^{\frac{1}% {2}\beta+\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 β | q ) ⁢ P n ( α , β ) ⁡ ( x | q ) ] = - 2 ⁢ q - 1 2 ⁢ α + 1 4 ⁢ ( 1 - q n + 1 ) ⁢ ( 1 + q 1 2 ⁢ ( α + β - 1 ) ) ⁢ ( 1 + q 1 2 ⁢ ( α + β ) ) 1 - q ⁢ w ~ ⁢ ( x ; q α - 1 , q β - 1 | q ) ⁢ P n + 1 ( α - 1 , β - 1 ) ⁢ ( x | q ) subscript 𝐷 𝑞 delimited-[] ~ 𝑤 𝑥 superscript 𝑞 𝛼 conditional superscript 𝑞 𝛽 𝑞 continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 2 superscript 𝑞 1 2 𝛼 1 4 1 superscript 𝑞 𝑛 1 1 superscript 𝑞 1 2 𝛼 𝛽 1 1 superscript 𝑞 1 2 𝛼 𝛽 1 𝑞 ~ 𝑤 𝑥 superscript 𝑞 𝛼 1 conditional superscript 𝑞 𝛽 1 𝑞 superscript subscript 𝑃 𝑛 1 𝛼 1 𝛽 1 conditional 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle D_{q}\left[{\tilde{w}}(x;q^{\alpha}% ,q^{\beta}|q)P^{(\alpha,\beta)}_{n}\!\left(x|q\right)\right]{}=-2q^{-\frac{1}{% 2}\alpha+\frac{1}{4}}\frac{(1-q^{n+1})(1+q^{\frac{1}{2}(\alpha+\beta-1)})(1+q^% {\frac{1}{2}(\alpha+\beta)})}{1-q}{}{\tilde{w}}(x;q^{\alpha-1},q^{\beta-1}|q)P% _{n+1}^{(\alpha-1,\beta-1)}(x|q)}}} {\displaystyle D_q\left[{\tilde w}(x;q^{\alpha},q^{\beta}|q)\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}\right] {}=-2q^{-\frac{1}{2}\alpha+\frac{1}{4}} \frac{(1-q^{n+1})(1+q^{\frac{1}{2}(\alpha+\beta-1)})(1+q^{\frac{1}{2}(\alpha+\beta)})}{1-q} {}{\tilde w}(x;q^{\alpha-1},q^{\beta-1}|q)P_{n+1}^{(\alpha-1,\beta-1)}(x|q) }

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

w ⁢ ( x ) := w ⁢ ( x ; q α , q β | 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 1 2 ⁢ β + 1 4 ⁢ e i ⁢ θ , - q 1 2 ⁢ β + 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 ⁢ β + 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 ) ⁢ h ⁢ ( x , - q 1 2 ⁢ β + 1 4 ) ⁢ h ⁢ ( x , - q 1 2 ⁢ β + 3 4 ) assign 𝑤 𝑥 𝑤 𝑥 superscript 𝑞 𝛼 conditional superscript 𝑞 𝛽 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 superscript 𝑞 1 2 𝛽 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛽 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 𝛽 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 ℎ 𝑥 superscript 𝑞 1 2 𝛽 1 4 ℎ 𝑥 superscript 𝑞 1 2 𝛽 3 4 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha},q^{\beta}|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^{\frac{1}{2}\beta+\frac{1% }{4}}{\mathrm{e}^{\mathrm{i}\theta}},-q^{\frac{1}{2}\beta+\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}\beta+\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,-q^{\frac{1}{2}\beta+\frac{1}{4}})h(x,-q^{\frac{1}% {2}\beta+\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 β | q ) ⁢ P n ( α , β ) ⁡ ( x | q ) = ( q - 1 2 ) n ⁢ q 1 4 ⁢ n 2 + 1 2 ⁢ n ⁢ α ( q , - q 1 2 ⁢ ( α + β + 1 ) , - q 1 2 ⁢ ( α + β + 2 ) ; q ) n ⁢ ( D q ) n ⁢ [ w ~ ⁢ ( x ; q α + n , q β + n | q ) ] ~ 𝑤 𝑥 superscript 𝑞 𝛼 conditional superscript 𝑞 𝛽 𝑞 continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 superscript 𝑞 1 2 𝑛 superscript 𝑞 1 4 superscript 𝑛 2 1 2 𝑛 𝛼 q-Pochhammer-symbol 𝑞 superscript 𝑞 1 2 𝛼 𝛽 1 superscript 𝑞 1 2 𝛼 𝛽 2 𝑞 𝑛 superscript subscript 𝐷 𝑞 𝑛 delimited-[] ~ 𝑤 𝑥 superscript 𝑞 𝛼 𝑛 conditional superscript 𝑞 𝛽 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;q^{\alpha},q^{\beta}|q% )P^{(\alpha,\beta)}_{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^{\frac{1}{2}(\alpha+% \beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)};q\right)_{n}}{}\left(D_{q}\right)^{% n}\left[{\tilde{w}}(x;q^{\alpha+n},q^{\beta+n}|q)\right]}}} {\displaystyle {\tilde w}(x;q^\alpha,q^\beta|q)\ctsqJacobi{\alpha}{\beta}{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^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{n}} {}\left(D_q\right)^n\left[{\tilde w}(x;q^{\alpha+n},q^{\beta+n}|q)\right] }

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

w ⁢ ( x ) := w ⁢ ( x ; q α , q β | 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 1 2 ⁢ β + 1 4 ⁢ e i ⁢ θ , - q 1 2 ⁢ β + 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 ⁢ β + 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 ) ⁢ h ⁢ ( x , - q 1 2 ⁢ β + 1 4 ) ⁢ h ⁢ ( x , - q 1 2 ⁢ β + 3 4 ) assign 𝑤 𝑥 𝑤 𝑥 superscript 𝑞 𝛼 conditional superscript 𝑞 𝛽 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 superscript 𝑞 1 2 𝛽 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛽 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 𝛽 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 ℎ 𝑥 superscript 𝑞 1 2 𝛽 1 4 ℎ 𝑥 superscript 𝑞 1 2 𝛽 3 4 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha},q^{\beta}|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^{\frac{1}{2}\beta+\frac{1% }{4}}{\mathrm{e}^{\mathrm{i}\theta}},-q^{\frac{1}{2}\beta+\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}\beta+\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,-q^{\frac{1}{2}\beta+\frac{1}{4}})h(x,-q^{\frac{1}% {2}\beta+\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

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


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


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


Limit relations

Askey-Wilson polynomial to Continuous q-Jacobi polynomial

q ( 1 2 ⁢ α + 1 4 ) ⁢ n ⁢ p n ⁡ ( x ; q 1 2 ⁢ α + 1 4 , q 1 2 ⁢ α + 3 4 , - q 1 2 ⁢ β + 1 4 , - q 1 2 ⁢ β + 3 4 | q ) ( q , - q 1 2 ⁢ ( α + β + 1 ) , - q 1 2 ⁢ ( α + β + 2 ) ; q ) n = P n ( α , β ) ⁡ ( x | q ) superscript 𝑞 1 2 𝛼 1 4 𝑛 Askey-Wilson-polynomial-p 𝑛 𝑥 superscript 𝑞 1 2 𝛼 1 4 superscript 𝑞 1 2 𝛼 3 4 superscript 𝑞 1 2 𝛽 1 4 superscript 𝑞 1 2 𝛽 3 4 𝑞 q-Pochhammer-symbol 𝑞 superscript 𝑞 1 2 𝛼 𝛽 1 superscript 𝑞 1 2 𝛼 𝛽 2 𝑞 𝑛 continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{q^{(\frac{1}{2}\alpha+\frac{1}% {4})n}p_{n}\!\left(x;q^{\frac{1}{2}\alpha+\frac{1}{4}},q^{\frac{1}{2}\alpha+% \frac{3}{4}},-q^{\frac{1}{2}\beta+\frac{1}{4}},-q^{\frac{1}{2}\beta+\frac{3}{4% }}\,|\,q\right)}{\left(q,-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(% \alpha+\beta+2)};q\right)_{n}}=P^{(\alpha,\beta)}_{n}\!\left(x|q\right)}}} {\displaystyle \frac{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}\AskeyWilson{n}@{x}{q^{\frac{1}{2}\alpha+\frac{1}{4}}}{q^{\frac{1}{2}\alpha+\frac{3}{4}}}{ -q^{\frac{1}{2}\beta+\frac{1}{4}}}{-q^{\frac{1}{2}\beta+\frac{3}{4}}}{q}} {\qPochhammer{q,-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{n}} =\ctsqJacobi{\alpha}{\beta}{n}@{x}{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-Jacobi polynomial to Jacobi polynomial

lim q → 1 ⁡ P n ( α , β ) ⁡ ( x | q ) = P n ( α , β ) ⁡ ( x ) subscript → 𝑞 1 continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}P^{(\alpha,% \beta)}_{n}\!\left(x|q\right)=P^{(\alpha,\beta)}_{n}\left(x\right)}}} {\displaystyle \lim_{q\rightarrow 1}\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}=\Jacobi{\alpha}{\beta}{n}@{x} }

Remarks

P n ( α , β ) ⁡ ( x ; q ) = ( q α + 1 , - q β + 1 ; q ) n ( q , - q ; q ) n ⁢ \qHyperrphis ⁢ 43 ⁢ @ ⁢ @ ⁢ q - n , q n + α + β + 1 , q 1 2 ⁢ e i ⁢ θ , q 1 2 ⁢ e - i ⁢ θ ⁢ q α + 1 , - q β + 1 , - q ⁢ q ⁢ q continuous-q-Jacobi-Rahman-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛼 1 superscript 𝑞 𝛽 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑞 𝑛 \qHyperrphis 43 @ @ superscript 𝑞 𝑛 superscript 𝑞 𝑛 𝛼 𝛽 1 superscript 𝑞 1 2 imaginary-unit 𝜃 superscript 𝑞 1 2 imaginary-unit 𝜃 superscript 𝑞 𝛼 1 superscript 𝑞 𝛽 1 𝑞 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\!\left(x;q% \right)=\frac{\left(q^{\alpha+1},-q^{\beta+1};q\right)_{n}}{\left(q,-q;q\right% )_{n}}{}\qHyperrphis{4}{3}@@{q^{-n},q^{n+\alpha+\beta+1},q^{\frac{1}{2}}{% \mathrm{e}^{\mathrm{i}\theta}},q^{\frac{1}{2}}{\mathrm{e}^{-\mathrm{i}\theta}}% }{q^{\alpha+1},-q^{\beta+1},-q}{q}{q}}}} {\displaystyle \ctsqJacobiRahman{\alpha}{\beta}{n}@{x}{q}=\frac{\qPochhammer{q^{\alpha+1},-q^{\beta+1}}{q}{n}}{\qPochhammer{q,-q}{q}{n}} {} \qHyperrphis{4}{3}@@{q^{-n},q^{n+\alpha+\beta+1},q^{\frac{1}{2}}\expe^{\iunit\theta},q^{\frac{1}{2}}\expe^{-\iunit\theta}}{q^{\alpha+1},-q^{\beta+1},-q}{q}{q} }
P n ( α , β ) ⁡ ( x | q 2 ) = ( - q ; q ) n ( - q α + β + 1 ; q ) n ⁢ q n ⁢ α ⁢ P n ( α , β ) ⁡ ( x ; q ) continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 𝑞 2 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝛼 𝛽 1 𝑞 𝑛 superscript 𝑞 𝑛 𝛼 continuous-q-Jacobi-Rahman-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\!\left(x|q^{% 2}\right)=\frac{\left(-q;q\right)_{n}}{\left(-q^{\alpha+\beta+1};q\right)_{n}}% q^{n\alpha}P^{(\alpha,\beta)}_{n}\!\left(x;q\right)}}} {\displaystyle \ctsqJacobi{\alpha}{\beta}{n}@{x}{q^2}=\frac{\qPochhammer{-q}{q}{n}}{\qPochhammer{-q^{\alpha+\beta+1}}{q}{n}}q^{n\alpha}\ctsqJacobiRahman{\alpha}{\beta}{n}@{x}{q} }
C 2 ⁢ n ⁡ ( x ; q λ | q ) = ( q λ , - q ; q ) n ( q 1 2 , - q 1 2 ; q ) n ⁢ q - 1 2 ⁢ n ⁢ P n ( λ - 1 2 , - 1 2 ) ⁡ ( 2 ⁢ x 2 - 1 ; q ) continuous-q-ultraspherical-Rogers-polynomial 2 𝑛 𝑥 superscript 𝑞 𝜆 𝑞 q-Pochhammer-symbol superscript 𝑞 𝜆 𝑞 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 1 2 superscript 𝑞 1 2 𝑞 𝑛 superscript 𝑞 1 2 𝑛 continuous-q-Jacobi-Rahman-polynomial-P 𝜆 1 2 1 2 𝑛 2 superscript 𝑥 2 1 𝑞 {\displaystyle{\displaystyle{\displaystyle C_{2n}\!\left(x;q^{\lambda}\,|\,q% \right)=\frac{\left(q^{\lambda},-q;q\right)_{n}}{\left(q^{\frac{1}{2}},-q^{% \frac{1}{2}};q\right)_{n}}q^{-\frac{1}{2}n}P^{(\lambda-\frac{1}{2},-\frac{1}{2% })}_{n}\!\left(2x^{2}-1;q\right)}}} {\displaystyle \ctsqUltra{2n}@{x}{q^{\lambda}}{q}=\frac{\qPochhammer{q^{\lambda},-q}{q}{n}} {\qPochhammer{q^{\frac{1}{2}},-q^{\frac{1}{2}}}{q}{n}}q^{-\frac{1}{2}n} \ctsqJacobiRahman{\lambda-\frac{1}{2}}{-\frac{1}{2}}{n}@{2x^2-1}{q} }
C 2 ⁢ n + 1 ⁡ ( x ; q λ | q ) = ( q λ , - 1 ; q ) n + 1 ( q 1 2 , - q 1 2 ; q ) n + 1 ⁢ q - 1 2 ⁢ n ⁢ x ⁢ P n ( λ - 1 2 , 1 2 ) ⁡ ( 2 ⁢ x 2 - 1 ; q ) continuous-q-ultraspherical-Rogers-polynomial 2 𝑛 1 𝑥 superscript 𝑞 𝜆 𝑞 q-Pochhammer-symbol superscript 𝑞 𝜆 1 𝑞 𝑛 1 q-Pochhammer-symbol superscript 𝑞 1 2 superscript 𝑞 1 2 𝑞 𝑛 1 superscript 𝑞 1 2 𝑛 𝑥 continuous-q-Jacobi-Rahman-polynomial-P 𝜆 1 2 1 2 𝑛 2 superscript 𝑥 2 1 𝑞 {\displaystyle{\displaystyle{\displaystyle C_{2n+1}\!\left(x;q^{\lambda}\,|\,q% \right)=\frac{\left(q^{\lambda},-1;q\right)_{n+1}}{\left(q^{\frac{1}{2}},-q^{% \frac{1}{2}};q\right)_{n+1}}q^{-\frac{1}{2}n}xP^{(\lambda-\frac{1}{2},\frac{1}% {2})}_{n}\!\left(2x^{2}-1;q\right)}}} {\displaystyle \ctsqUltra{2n+1}@{x}{q^{\lambda}}{q}=\frac{\qPochhammer{q^{\lambda},-1}{q}{n+1}} {\qPochhammer{q^{\frac{1}{2}},-q^{\frac{1}{2}}}{q}{n+1}}q^{-\frac{1}{2}n} x\ctsqJacobiRahman{\lambda-\frac{1}{2}}{\frac{1}{2}}{n}@{2x^2-1}{q} }
P n ( α , β ) ⁡ ( x | q - 1 ) = q - n ⁢ α ⁢ P n ( α , β ) ⁡ ( x | q )   and   P n ( α , β ) ⁡ ( x ; q - 1 ) = q - n ⁢ ( α + β ) ⁢ P n ( α , β ) ⁡ ( x ; q ) formulae-sequence continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 𝑞 1 superscript 𝑞 𝑛 𝛼 continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 and continuous-q-Jacobi-Rahman-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 𝑞 1 superscript 𝑞 𝑛 𝛼 𝛽 continuous-q-Jacobi-Rahman-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\!\left(x|q^{% -1}\right)=q^{-n\alpha}P^{(\alpha,\beta)}_{n}\!\left(x|q\right)\quad\textrm{% and}\quad P^{(\alpha,\beta)}_{n}\!\left(x;q^{-1}\right)=q^{-n(\alpha+\beta)}P^% {(\alpha,\beta)}_{n}\!\left(x;q\right)}}} {\displaystyle \ctsqJacobi{\alpha}{\beta}{n}@{x}{q^{-1}}=q^{-n\alpha}\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}\quad\textrm{and}\quad \ctsqJacobiRahman{\alpha}{\beta}{n}@{x}{q^{-1}}=q^{-n(\alpha+\beta)}\ctsqJacobiRahman{\alpha}{\beta}{n}@{x}{q} }

Koornwinder Addendum: q-Meixner-Pollaczek

P n ⁡ ( x ; a | q ) := 1 ( q ; q ) n p n ⁡ ( x ; a ⁢ e i ⁢ ϕ , 0 , a ⁢ e - i ⁢ ϕ , 0 | q ) ( x = cos ⁡ ( θ + ϕ ) ) fragments q-Meixner-Pollaczek-polynomial-P 𝑛 𝑥 𝑎 𝑞 assign 1 q-Pochhammer-symbol 𝑞 𝑞 𝑛 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 imaginary-unit italic-ϕ 0 𝑎 imaginary-unit italic-ϕ 0 𝑞 fragments ( x 𝜃 italic-ϕ ) {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a|q\right):=\frac{1}% {\left(q;q\right)_{n}}p_{n}\!\left(x;a{\mathrm{e}^{\mathrm{i}\phi}},0,a{% \mathrm{e}^{-\mathrm{i}\phi}},0\,|\,q\right)(x=\cos\left(\theta+\phi\right))}}} {\displaystyle \qMeixnerPollaczek{n}@{x}{a }{ q}:=\frac1{\qPochhammer{q}{q}{n}} \AskeyWilson{n}@{x}{a \expe^{\iunit\phi}}{0}{a \expe^{-\iunit\phi}}{0 }{ q} (x=\cos@{\theta+\phi}) }

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