Continuous q-Jacobi: Special cases

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Continuous q-Jacobi: Special cases

Continuous q-ultraspherical / Rogers

Basic hypergeometric representation

C n ⁡ ( x ; β | q ) = ( β 2 ; q ) n ( q ; q ) n ⁢ β - 1 2 ⁢ n ⁢ \qHyperrphis ⁢ 43 ⁢ @ ⁢ @ ⁢ q - n , β 2 ⁢ q n ⁢ β 1 2 ⁢ e i ⁢ θ , β 1 2 ⁢ e - i ⁢ θ ⁢ β ⁢ q 1 2 - β , - β ⁢ q 1 2 ⁢ q ⁢ q continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 q-Pochhammer-symbol superscript 𝛽 2 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝛽 1 2 𝑛 \qHyperrphis 43 @ @ superscript 𝑞 𝑛 superscript 𝛽 2 superscript 𝑞 𝑛 superscript 𝛽 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 imaginary-unit 𝜃 𝛽 superscript 𝑞 1 2 𝛽 𝛽 superscript 𝑞 1 2 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;\beta\,|\,q\right)=% \frac{\left(\beta^{2};q\right)_{n}}{\left(q;q\right)_{n}}\beta^{-\frac{1}{2}n}% \,\qHyperrphis{4}{3}@@{q^{-n},\beta^{2}q^{n}\beta^{\frac{1}{2}}{\mathrm{e}^{% \mathrm{i}\theta}},\beta^{\frac{1}{2}}{\mathrm{e}^{-\mathrm{i}\theta}}}{\beta q% ^{\frac{1}{2}}-\beta,-\beta q^{\frac{1}{2}}}{q}{q}}}} {\displaystyle \ctsqUltra{n}@{x}{\beta}{q}=\frac{\qPochhammer{\beta^2}{q}{n}}{\qPochhammer{q}{q}{n}}\beta^{-\frac{1}{2}n}\,\qHyperrphis{4}{3}@@{q^{-n},\beta^2q^n \beta^{\frac{1}{2}}\expe^{\iunit\theta},\beta^{\frac{1}{2}}\expe^{-\iunit\theta}}{\beta q^{\frac{1}{2}} -\beta,-\beta q^{\frac{1}{2}}}{q}{q} }

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


C n ⁡ ( x ; β | q ) = ( β 2 ; q ) n ( q ; q ) n ⁢ β - n ⁢ e - i ⁢ n ⁢ θ ⁢ \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ q - n , β , β ⁢ e 2 ⁢ i ⁢ θ ⁢ β 2 , 0 ⁢ q ⁢ q continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 q-Pochhammer-symbol superscript 𝛽 2 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝛽 𝑛 imaginary-unit 𝑛 𝜃 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 𝛽 𝛽 2 imaginary-unit 𝜃 superscript 𝛽 2 0 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;\beta\,|\,q\right)=% \frac{\left(\beta^{2};q\right)_{n}}{\left(q;q\right)_{n}}\beta^{-n}{\mathrm{e}% ^{-\mathrm{i}n\theta}}\,\qHyperrphis{3}{2}@@{q^{-n},\beta,\beta{\mathrm{e}^{2% \mathrm{i}\theta}}}{\beta^{2},0}{q}{q}}}} {\displaystyle \ctsqUltra{n}@{x}{\beta}{q}=\frac{\qPochhammer{\beta^2}{q}{n}}{\qPochhammer{q}{q}{n}}\beta^{-n}\expe^{-\iunit n\theta}\,\qHyperrphis{3}{2}@@{q^{-n},\beta,\beta\expe^{2\iunit\theta}}{\beta^2,0}{q}{q} }
C n ⁡ ( x ; β | q ) = ( β ; q ) n ( q ; q ) n ⁢ e i ⁢ n ⁢ θ ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - n , β ⁢ β - 1 ⁢ q - n + 1 ⁢ q ⁢ β - 1 ⁢ q ⁢ e - 2 ⁢ i ⁢ θ continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 q-Pochhammer-symbol 𝛽 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 imaginary-unit 𝑛 𝜃 \qHyperrphis 21 @ @ superscript 𝑞 𝑛 𝛽 superscript 𝛽 1 superscript 𝑞 𝑛 1 𝑞 superscript 𝛽 1 𝑞 2 imaginary-unit 𝜃 {\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;\beta\,|\,q\right)=% \frac{\left(\beta;q\right)_{n}}{\left(q;q\right)_{n}}{\mathrm{e}^{\mathrm{i}n% \theta}}\,\qHyperrphis{2}{1}@@{q^{-n},\beta}{\beta^{-1}q^{-n+1}}{q}{\beta^{-1}% q{\mathrm{e}^{-2\mathrm{i}\theta}}}}}} {\displaystyle \ctsqUltra{n}@{x}{\beta}{q}=\frac{\qPochhammer{\beta}{q}{n}}{\qPochhammer{q}{q}{n}}\expe^{\iunit n\theta}\,\qHyperrphis{2}{1}@@{q^{-n},\beta}{\beta^{-1}q^{-n+1}}{q}{\beta^{-1}q\expe^{-2\iunit\theta}} }

Orthogonality relation(s)

1 2 ⁢ π ⁢ ∫ - 1 1 w ⁢ ( x ) 1 - x 2 ⁢ C m ⁡ ( x ; β | q ) ⁢ C n ⁡ ( x ; β | q ) ⁢ 𝑑 x = ( β , β ⁢ q ; q ) ∞ ( β 2 , q ; q ) ∞ ⁢ ( β 2 ; q ) n ( q ; q ) n ⁢ ( 1 - β ) ( 1 - β ⁢ q n ) ⁢ δ m , n 1 2 superscript subscript 1 1 𝑤 𝑥 1 superscript 𝑥 2 continuous-q-ultraspherical-Rogers-polynomial 𝑚 𝑥 𝛽 𝑞 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 differential-d 𝑥 q-Pochhammer-symbol 𝛽 𝛽 𝑞 𝑞 q-Pochhammer-symbol superscript 𝛽 2 𝑞 𝑞 q-Pochhammer-symbol superscript 𝛽 2 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 1 𝛽 1 𝛽 superscript 𝑞 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x% )}{\sqrt{1-x^{2}}}C_{m}\!\left(x;\beta\,|\,q\right)C_{n}\!\left(x;\beta\,|\,q% \right)\,dx{}=\frac{\left(\beta,\beta q;q\right)_{\infty}}{\left(\beta^{2},q;q% \right)_{\infty}}\frac{\left(\beta^{2};q\right)_{n}}{\left(q;q\right)_{n}}% \frac{(1-\beta)}{(1-\beta q^{n})}\,\delta_{m,n}}}} {\displaystyle \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\ctsqUltra{m}@{x}{\beta}{q}\ctsqUltra{n}@{x}{\beta}{q}\,dx {}=\frac{\qPochhammer{\beta,\beta q}{q}{\infty}}{\qPochhammer{\beta^2,q}{q}{\infty}} \frac{\qPochhammer{\beta^2}{q}{n}}{\qPochhammer{q}{q}{n}}\frac{(1-\beta)}{(1-\beta q^n)}\,\Kronecker{m}{n} }

Constraint(s): | β | < 1 𝛽 1 {\displaystyle{\displaystyle{\displaystyle|\beta|<1}}}


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

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 ⁢ ( 1 - β ⁢ q n ) ⁢ x ⁢ C n ⁡ ( x ; β | q ) = ( 1 - q n + 1 ) ⁢ C n + 1 ⁡ ( x ; β | q ) + ( 1 - β 2 ⁢ q n - 1 ) ⁢ C n - 1 ⁡ ( x ; β | q ) 2 1 𝛽 superscript 𝑞 𝑛 𝑥 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 1 superscript 𝑞 𝑛 1 continuous-q-ultraspherical-Rogers-polynomial 𝑛 1 𝑥 𝛽 𝑞 1 superscript 𝛽 2 superscript 𝑞 𝑛 1 continuous-q-ultraspherical-Rogers-polynomial 𝑛 1 𝑥 𝛽 𝑞 {\displaystyle{\displaystyle{\displaystyle 2(1-\beta q^{n})xC_{n}\!\left(x;% \beta\,|\,q\right)=(1-q^{n+1})C_{n+1}\!\left(x;\beta\,|\,q\right){}+(1-\beta^{% 2}q^{n-1})C_{n-1}\!\left(x;\beta\,|\,q\right)}}} {\displaystyle 2(1-\beta q^n)x\ctsqUltra{n}@{x}{\beta}{q}=(1-q^{n+1})\ctsqUltra{n+1}@{x}{\beta}{q} {}+(1-\beta^2 q^{n-1})\ctsqUltra{n-1}@{x}{\beta}{q} }

Monic recurrence relation

x ⁢ C ^ n ⁡ ( x ) = C ^ n + 1 ⁡ ( x ) + ( 1 - q n ) ⁢ ( 1 - β 2 ⁢ q n - 1 ) 4 ⁢ ( 1 - β ⁢ q n - 1 ) ⁢ ( 1 - β ⁢ q n ) ⁢ C ^ n - 1 ⁡ ( x ) 𝑥 continuous-q-ultraspherical-Rogers-polynomial-monic-p 𝑛 𝑥 𝛽 𝑞 continuous-q-ultraspherical-Rogers-polynomial-monic-p 𝑛 1 𝑥 𝛽 𝑞 1 superscript 𝑞 𝑛 1 superscript 𝛽 2 superscript 𝑞 𝑛 1 4 1 𝛽 superscript 𝑞 𝑛 1 1 𝛽 superscript 𝑞 𝑛 continuous-q-ultraspherical-Rogers-polynomial-monic-p 𝑛 1 𝑥 𝛽 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{C}}_{n}\!\left(x\right)=% {\widehat{C}}_{n+1}\!\left(x\right)+\frac{(1-q^{n})(1-\beta^{2}q^{n-1})}{4(1-% \beta q^{n-1})(1-\beta q^{n})}{\widehat{C}}_{n-1}\!\left(x\right)}}} {\displaystyle x\monicctsqUltra{n}@@{x}{\beta}{q}=\monicctsqUltra{n+1}@@{x}{\beta}{q}+ \frac{(1-q^n)(1-\beta^2q^{n-1})}{4(1-\beta q^{n-1})(1-\beta q^n)}\monicctsqUltra{n-1}@@{x}{\beta}{q} }
C n ⁡ ( x ; β | q ) = 2 n ⁢ ( β ; q ) n ( q ; q ) n ⁢ C ^ n ⁡ ( x ) continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 superscript 2 𝑛 q-Pochhammer-symbol 𝛽 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 continuous-q-ultraspherical-Rogers-polynomial-monic-p 𝑛 𝑥 𝛽 𝑞 {\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;\beta\,|\,q\right)=% \frac{2^{n}\left(\beta;q\right)_{n}}{\left(q;q\right)_{n}}{\widehat{C}}_{n}\!% \left(x\right)}}} {\displaystyle \ctsqUltra{n}@{x}{\beta}{q}=\frac{2^n\qPochhammer{\beta}{q}{n}}{\qPochhammer{q}{q}{n}}\monicctsqUltra{n}@@{x}{\beta}{q} }

q-Difference equation

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

Substitution(s): λ n = 4 ⁢ q - n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 - β 2 ⁢ q n ) subscript 𝜆 𝑛 4 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 superscript 𝛽 2 superscript 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\lambda_{n}=4q^{-n+1}(1-q^{n})(1-% \beta^{2}q^{n})}}} &

w ~ ⁢ ( x ; β | q ) := w ⁢ ( x ; β | q ) 1 - x 2 assign ~ 𝑤 𝑥 conditional 𝛽 𝑞 𝑤 𝑥 conditional 𝛽 𝑞 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;\beta|q):=\frac{w(x;% \beta|q)}{\sqrt{1-x^{2}}}}}} &
y ⁢ ( x ) = C n ⁡ ( x ; β | q ) 𝑦 𝑥 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=C_{n}\!\left(x;\beta\,|\,q% \right)}}} &
w ⁢ ( x ) := w ⁢ ( x ; β | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( β 1 2 ⁢ e i ⁢ θ , β 1 2 ⁢ q 1 2 ⁢ e i ⁢ θ - β 1 2 ⁢ e i ⁢ θ , - β 1 2 ⁢ q 1 2 ⁢ e i ⁢ θ ; q ) ∞ | 2 = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( β ⁢ e 2 ⁢ i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , β 1 2 ) ⁢ h ⁢ ( x , β 1 2 ⁢ q 1 2 ) ⁢ h ⁢ ( x , - β 1 2 ) ⁢ h ⁢ ( x , - β 1 2 ⁢ q 1 2 ) assign 𝑤 𝑥 𝑤 𝑥 conditional 𝛽 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol superscript 𝛽 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 superscript 𝑞 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 superscript 𝑞 1 2 imaginary-unit 𝜃 𝑞 2 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝛽 2 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 superscript 𝛽 1 2 ℎ 𝑥 superscript 𝛽 1 2 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝛽 1 2 ℎ 𝑥 superscript 𝛽 1 2 superscript 𝑞 1 2 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;\beta|q)=\left|\frac{% \left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(\beta^{\frac{1% }{2}}{\mathrm{e}^{\mathrm{i}\theta}},\beta^{\frac{1}{2}}q^{\frac{1}{2}}{% \mathrm{e}^{\mathrm{i}\theta}}-\beta^{\frac{1}{2}}{\mathrm{e}^{\mathrm{i}% \theta}},-\beta^{\frac{1}{2}}q^{\frac{1}{2}}{\mathrm{e}^{\mathrm{i}\theta}};q% \right)_{\infty}}\right|^{2}=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}% };q\right)_{\infty}}{\left(\beta{\mathrm{e}^{2\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,\beta^{\frac{1}{2}})h(x,\beta^{\frac{1}{2}}q^{\frac{1}{2}})h(x,-\beta^% {\frac{1}{2}})h(x,-\beta^{\frac{1}{2}}q^{\frac{1}{2}})}}}} &
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 ⁢ C n ⁡ ( x ; β | q ) = - q - 1 2 ⁢ n ⁢ ( 1 - β ) ⁢ ( e i ⁢ θ - e - i ⁢ θ ) ⁢ C n - 1 ⁡ ( x ; β ⁢ q | q ) subscript 𝛿 𝑞 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 superscript 𝑞 1 2 𝑛 1 𝛽 imaginary-unit 𝜃 imaginary-unit 𝜃 continuous-q-ultraspherical-Rogers-polynomial 𝑛 1 𝑥 𝛽 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}C_{n}\!\left(x;\beta\,|\,q% \right)=-q^{-\frac{1}{2}n}(1-\beta)({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e% }^{-\mathrm{i}\theta}})C_{n-1}\!\left(x;\beta q\,|\,q\right)}}} {\displaystyle \delta_q\ctsqUltra{n}@{x}{\beta}{q}=-q^{-\frac{1}{2}n}(1-\beta)(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) \ctsqUltra{n-1}@{x}{\beta q}{q} }

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


D q ⁢ C n ⁡ ( x ; β | q ) = 2 ⁢ q - 1 2 ⁢ ( n - 1 ) ⁢ 1 - β 1 - q ⁢ C n - 1 ⁡ ( x ; β ⁢ q | q ) subscript 𝐷 𝑞 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 2 superscript 𝑞 1 2 𝑛 1 1 𝛽 1 𝑞 continuous-q-ultraspherical-Rogers-polynomial 𝑛 1 𝑥 𝛽 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle D_{q}C_{n}\!\left(x;\beta\,|\,q% \right)=2q^{-\frac{1}{2}(n-1)}\frac{1-\beta}{1-q}C_{n-1}\!\left(x;\beta q\,|\,% q\right)}}} {\displaystyle D_q\ctsqUltra{n}@{x}{\beta}{q}=2q^{-\frac{1}{2}(n-1)}\frac{1-\beta}{1-q}\ctsqUltra{n-1}@{x}{\beta q}{q} }

Backward shift operator

δ q ⁢ [ w ~ ⁢ ( x ; β | q ) ⁢ C n ⁡ ( x ; β | q ) ] = q - 1 2 ⁢ ( n + 1 ) ⁢ ( 1 - q n + 1 ) ⁢ ( 1 - β 2 ⁢ q n - 1 ) ( 1 - β ⁢ q - 1 ) ⁢ ( e i ⁢ θ - e - i ⁢ θ ) ⁢ w ~ ⁢ ( x ; β ⁢ q - 1 | q ) ⁢ C n + 1 ⁡ ( x ; β ⁢ q - 1 | q ) subscript 𝛿 𝑞 delimited-[] ~ 𝑤 𝑥 conditional 𝛽 𝑞 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 superscript 𝑞 1 2 𝑛 1 1 superscript 𝑞 𝑛 1 1 superscript 𝛽 2 superscript 𝑞 𝑛 1 1 𝛽 superscript 𝑞 1 imaginary-unit 𝜃 imaginary-unit 𝜃 ~ 𝑤 𝑥 conditional 𝛽 superscript 𝑞 1 𝑞 continuous-q-ultraspherical-Rogers-polynomial 𝑛 1 𝑥 𝛽 superscript 𝑞 1 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}\left[{\tilde{w}}(x;\beta|% q)C_{n}\!\left(x;\beta\,|\,q\right)\right]{}=q^{-\frac{1}{2}(n+1)}\frac{(1-q^{% n+1})(1-\beta^{2}q^{n-1})}{(1-\beta q^{-1})}({\mathrm{e}^{\mathrm{i}\theta}}-{% \mathrm{e}^{-\mathrm{i}\theta}}){}{\tilde{w}}(x;\beta q^{-1}|q)C_{n+1}\!\left(% x;\beta q^{-1}\,|\,q\right)}}} {\displaystyle \delta_q\left[{\tilde w}(x;\beta|q)\ctsqUltra{n}@{x}{\beta}{q}\right] {}=q^{-\frac{1}{2}(n+1)}\frac{(1-q^{n+1})(1-\beta^2q^{n-1})}{(1-\beta q^{-1})} (\expe^{\iunit\theta}-\expe^{-\iunit\theta}) {}{\tilde w}(x;\beta q^{-1}|q)\ctsqUltra{n+1}@{x}{\beta q^{-1}}{q} }

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

w ⁢ ( x ) := w ⁢ ( x ; β | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( β 1 2 ⁢ e i ⁢ θ , β 1 2 ⁢ q 1 2 ⁢ e i ⁢ θ - β 1 2 ⁢ e i ⁢ θ , - β 1 2 ⁢ q 1 2 ⁢ e i ⁢ θ ; q ) ∞ | 2 = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( β ⁢ e 2 ⁢ i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , β 1 2 ) ⁢ h ⁢ ( x , β 1 2 ⁢ q 1 2 ) ⁢ h ⁢ ( x , - β 1 2 ) ⁢ h ⁢ ( x , - β 1 2 ⁢ q 1 2 ) assign 𝑤 𝑥 𝑤 𝑥 conditional 𝛽 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol superscript 𝛽 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 superscript 𝑞 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 superscript 𝑞 1 2 imaginary-unit 𝜃 𝑞 2 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝛽 2 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 superscript 𝛽 1 2 ℎ 𝑥 superscript 𝛽 1 2 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝛽 1 2 ℎ 𝑥 superscript 𝛽 1 2 superscript 𝑞 1 2 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;\beta|q)=\left|\frac{% \left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(\beta^{\frac{1% }{2}}{\mathrm{e}^{\mathrm{i}\theta}},\beta^{\frac{1}{2}}q^{\frac{1}{2}}{% \mathrm{e}^{\mathrm{i}\theta}}-\beta^{\frac{1}{2}}{\mathrm{e}^{\mathrm{i}% \theta}},-\beta^{\frac{1}{2}}q^{\frac{1}{2}}{\mathrm{e}^{\mathrm{i}\theta}};q% \right)_{\infty}}\right|^{2}=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}% };q\right)_{\infty}}{\left(\beta{\mathrm{e}^{2\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,\beta^{\frac{1}{2}})h(x,\beta^{\frac{1}{2}}q^{\frac{1}{2}})h(x,-\beta^% {\frac{1}{2}})h(x,-\beta^{\frac{1}{2}}q^{\frac{1}{2}})}}}} &
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 ; β | q ) ⁢ C n ⁡ ( x ; β | q ) ] = - 2 ⁢ q - 1 2 ⁢ n ⁢ ( 1 - q n + 1 ) ⁢ ( 1 - β 2 ⁢ q n - 1 ) ( 1 - q ) ⁢ ( 1 - β ⁢ q - 1 ) ⁢ w ~ ⁢ ( x ; β ⁢ q - 1 | q ) ⁢ C n + 1 ⁡ ( x ; β ⁢ q - 1 | q ) subscript 𝐷 𝑞 delimited-[] ~ 𝑤 𝑥 conditional 𝛽 𝑞 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 2 superscript 𝑞 1 2 𝑛 1 superscript 𝑞 𝑛 1 1 superscript 𝛽 2 superscript 𝑞 𝑛 1 1 𝑞 1 𝛽 superscript 𝑞 1 ~ 𝑤 𝑥 conditional 𝛽 superscript 𝑞 1 𝑞 continuous-q-ultraspherical-Rogers-polynomial 𝑛 1 𝑥 𝛽 superscript 𝑞 1 𝑞 {\displaystyle{\displaystyle{\displaystyle D_{q}\left[{\tilde{w}}(x;\beta|q)C_% {n}\!\left(x;\beta\,|\,q\right)\right]{}=-\frac{2q^{-\frac{1}{2}n}(1-q^{n+1})(% 1-\beta^{2}q^{n-1})}{(1-q)(1-\beta q^{-1})}{}{\tilde{w}}(x;\beta q^{-1}|q)C_{n% +1}\!\left(x;\beta q^{-1}\,|\,q\right)}}} {\displaystyle D_q\left[{\tilde w}(x;\beta|q)\ctsqUltra{n}@{x}{\beta}{q}\right] {}=-\frac{2q^{-\frac{1}{2}n}(1-q^{n+1})(1-\beta^2q^{n-1})}{(1-q)(1-\beta q^{-1})} {}{\tilde w}(x;\beta q^{-1}|q)\ctsqUltra{n+1}@{x}{\beta q^{-1}}{q} }

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

w ⁢ ( x ) := w ⁢ ( x ; β | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( β 1 2 ⁢ e i ⁢ θ , β 1 2 ⁢ q 1 2 ⁢ e i ⁢ θ - β 1 2 ⁢ e i ⁢ θ , - β 1 2 ⁢ q 1 2 ⁢ e i ⁢ θ ; q ) ∞ | 2 = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( β ⁢ e 2 ⁢ i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , β 1 2 ) ⁢ h ⁢ ( x , β 1 2 ⁢ q 1 2 ) ⁢ h ⁢ ( x , - β 1 2 ) ⁢ h ⁢ ( x , - β 1 2 ⁢ q 1 2 ) assign 𝑤 𝑥 𝑤 𝑥 conditional 𝛽 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol superscript 𝛽 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 superscript 𝑞 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 superscript 𝑞 1 2 imaginary-unit 𝜃 𝑞 2 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝛽 2 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 superscript 𝛽 1 2 ℎ 𝑥 superscript 𝛽 1 2 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝛽 1 2 ℎ 𝑥 superscript 𝛽 1 2 superscript 𝑞 1 2 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;\beta|q)=\left|\frac{% \left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(\beta^{\frac{1% }{2}}{\mathrm{e}^{\mathrm{i}\theta}},\beta^{\frac{1}{2}}q^{\frac{1}{2}}{% \mathrm{e}^{\mathrm{i}\theta}}-\beta^{\frac{1}{2}}{\mathrm{e}^{\mathrm{i}% \theta}},-\beta^{\frac{1}{2}}q^{\frac{1}{2}}{\mathrm{e}^{\mathrm{i}\theta}};q% \right)_{\infty}}\right|^{2}=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}% };q\right)_{\infty}}{\left(\beta{\mathrm{e}^{2\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,\beta^{\frac{1}{2}})h(x,\beta^{\frac{1}{2}}q^{\frac{1}{2}})h(x,-\beta^% {\frac{1}{2}})h(x,-\beta^{\frac{1}{2}}q^{\frac{1}{2}})}}}} &
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 ; β | q ) ⁢ C n ⁡ ( x ; β | q ) = ( q - 1 2 ) n ⁢ q 1 4 ⁢ n ⁢ ( n - 1 ) ⁢ ( β ; q ) n ( q , β 2 ⁢ q n ; q ) n ⁢ ( D q ) n ⁢ [ w ~ ⁢ ( x ; β ⁢ q n | q ) ] ~ 𝑤 𝑥 conditional 𝛽 𝑞 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 superscript 𝑞 1 2 𝑛 superscript 𝑞 1 4 𝑛 𝑛 1 q-Pochhammer-symbol 𝛽 𝑞 𝑛 q-Pochhammer-symbol 𝑞 superscript 𝛽 2 superscript 𝑞 𝑛 𝑞 𝑛 superscript subscript 𝐷 𝑞 𝑛 delimited-[] ~ 𝑤 𝑥 conditional 𝛽 superscript 𝑞 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;\beta|q)C_{n}\!\left(x% ;\beta\,|\,q\right){}=\left(\frac{q-1}{2}\right)^{n}q^{\frac{1}{4}n(n-1)}\frac% {\left(\beta;q\right)_{n}}{\left(q,\beta^{2}q^{n};q\right)_{n}}\left(D_{q}% \right)^{n}\left[{\tilde{w}}(x;\beta q^{n}|q)\right]}}} {\displaystyle {\tilde w}(x;\beta|q)\ctsqUltra{n}@{x}{\beta}{q} {}=\left(\frac{q-1}{2}\right)^n q^{\frac{1}{4}n(n-1)}\frac{\qPochhammer{\beta}{q}{n}}{\qPochhammer{q,\beta^2q^n}{q}{n}} \left(D_q\right)^n\left[{\tilde w}(x;\beta q^n|q)\right] }

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

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

| ( β ⁢ e i ⁢ θ ⁢ t ; q ) ∞ ( e i ⁢ θ ⁢ t ; q ) ∞ | 2 = ( β ⁢ e i ⁢ θ ⁢ t , β ⁢ e - i ⁢ θ ⁢ t ; q ) ∞ ( e i ⁢ θ ⁢ t , e - i ⁢ θ ⁢ t ; q ) ∞ = ∑ n = 0 ∞ C n ⁡ ( x ; β | q ) ⁢ t n superscript q-Pochhammer-symbol 𝛽 imaginary-unit 𝜃 𝑡 𝑞 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 𝑞 2 q-Pochhammer-symbol 𝛽 imaginary-unit 𝜃 𝑡 𝛽 imaginary-unit 𝜃 𝑡 𝑞 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 imaginary-unit 𝜃 𝑡 𝑞 superscript subscript 𝑛 0 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left|\frac{\left(\beta{\mathrm{e}^{% \mathrm{i}\theta}}t;q\right)_{\infty}}{\left({\mathrm{e}^{\mathrm{i}\theta}}t;% q\right)_{\infty}}\right|^{2}=\frac{\left(\beta{\mathrm{e}^{\mathrm{i}\theta}}% t,\beta{\mathrm{e}^{-\mathrm{i}\theta}}t;q\right)_{\infty}}{\left({\mathrm{e}^% {\mathrm{i}\theta}}t,{\mathrm{e}^{-\mathrm{i}\theta}}t;q\right)_{\infty}}=\sum% _{n=0}^{\infty}C_{n}\!\left(x;\beta\,|\,q\right)t^{n}}}} {\displaystyle \left|\frac{\qPochhammer{\beta\expe^{\iunit\theta}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\right|^2 =\frac{\qPochhammer{\beta\expe^{\iunit\theta}t,\beta\expe^{-\iunit\theta}t}{q}{\infty}} {\qPochhammer{\expe^{\iunit\theta}t,\expe^{-\iunit\theta}t}{q}{\infty}} =\sum_{n=0}^{\infty}\ctsqUltra{n}@{x}{\beta}{q}t^n }

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


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

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


( e - i ⁢ θ ⁢ t ; q ) ∞ ⋅ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ β , β ⁢ e 2 ⁢ i ⁢ θ ⁢ β 2 ⁢ q ⁢ e - i ⁢ θ ⁢ t = ∑ n = 0 ∞ ( - 1 ) n ⁢ β n ⁢ q \binomial ⁢ n ⁢ 2 ( β 2 ; q ) n ⁢ C n ⁡ ( x ; β | q ) ⁢ t n ⋅ q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 𝑞 \qHyperrphis 21 @ @ 𝛽 𝛽 2 imaginary-unit 𝜃 superscript 𝛽 2 𝑞 imaginary-unit 𝜃 𝑡 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝛽 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Pochhammer-symbol superscript 𝛽 2 𝑞 𝑛 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left({\mathrm{e}^{-\mathrm{i}\theta% }}t;q\right)_{\infty}\cdot\qHyperrphis{2}{1}@@{\beta,\beta{\mathrm{e}^{2% \mathrm{i}\theta}}}{\beta^{2}}{q}{{\mathrm{e}^{-\mathrm{i}\theta}}t}{}=\sum_{n% =0}^{\infty}\frac{(-1)^{n}\beta^{n}q^{\binomial{n}{2}}}{\left(\beta^{2};q% \right)_{n}}C_{n}\!\left(x;\beta\,|\,q\right)t^{n}}}} {\displaystyle \qPochhammer{\expe^{-\iunit\theta}t}{q}{\infty}\cdot \qHyperrphis{2}{1}@@{\beta,\beta\expe^{2\iunit\theta}}{\beta^2}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{(-1)^n\beta^nq^{\binomial{n}{2}}}{\qPochhammer{\beta^2}{q}{n}}\ctsqUltra{n}@{x}{\beta}{q}t^n }

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


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

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


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

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


\qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ β 1 2 ⁢ e i ⁢ θ - β 1 2 ⁢ q 1 2 ⁢ e i ⁢ θ - β ⁢ q 1 2 ⁢ q ⁢ e - i ⁢ θ ⁢ t ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ β 1 2 ⁢ q 1 2 ⁢ e - i ⁢ θ - β 1 2 ⁢ e - i ⁢ θ - β ⁢ q 1 2 ⁢ q ⁢ e i ⁢ θ ⁢ t = ∑ n = 0 ∞ ( - β , β ⁢ q 1 2 ; q ) n ( β 2 , - β ⁢ q 1 2 ; q ) n ⁢ C n ⁡ ( x ; β | q ) ⁢ t n \qHyperrphis 21 @ @ superscript 𝛽 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 superscript 𝑞 1 2 imaginary-unit 𝜃 𝛽 superscript 𝑞 1 2 𝑞 imaginary-unit 𝜃 𝑡 \qHyperrphis 21 @ @ superscript 𝛽 1 2 superscript 𝑞 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 imaginary-unit 𝜃 𝛽 superscript 𝑞 1 2 𝑞 imaginary-unit 𝜃 𝑡 superscript subscript 𝑛 0 q-Pochhammer-symbol 𝛽 𝛽 superscript 𝑞 1 2 𝑞 𝑛 q-Pochhammer-symbol superscript 𝛽 2 𝛽 superscript 𝑞 1 2 𝑞 𝑛 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{\beta^{\frac{1}% {2}}{\mathrm{e}^{\mathrm{i}\theta}}-\beta^{\frac{1}{2}}q^{\frac{1}{2}}{\mathrm% {e}^{\mathrm{i}\theta}}}{-\beta q^{\frac{1}{2}}}{q}{{\mathrm{e}^{-\mathrm{i}% \theta}}t}\ \qHyperrphis{2}{1}@@{\beta^{\frac{1}{2}}q^{\frac{1}{2}}{\mathrm{e}% ^{-\mathrm{i}\theta}}-\beta^{\frac{1}{2}}{\mathrm{e}^{-\mathrm{i}\theta}}}{-% \beta q^{\frac{1}{2}}}{q}{{\mathrm{e}^{\mathrm{i}\theta}}t}{}=\sum_{n=0}^{% \infty}\frac{\left(-\beta,\beta q^{\frac{1}{2}};q\right)_{n}}{\left(\beta^{2},% -\beta q^{\frac{1}{2}};q\right)_{n}}C_{n}\!\left(x;\beta\,|\,q\right)t^{n}}}} {\displaystyle \qHyperrphis{2}{1}@@{\beta^{\frac{1}{2}}\expe^{\iunit\theta} -\beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{\iunit\theta}}{-\beta q^{\frac{1}{2}}}{q}{\expe^{-\iunit\theta}t}\ \qHyperrphis{2}{1}@@{\beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{-\iunit\theta} -\beta^{\frac{1}{2}}\expe^{-\iunit\theta}}{-\beta q^{\frac{1}{2}}}{q}{\expe^{\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{-\beta,\beta q^{\frac{1}{2}}}{q}{n}} {\qPochhammer{\beta^2,-\beta q^{\frac{1}{2}}}{q}{n}}\ctsqUltra{n}@{x}{\beta}{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 , β , β ⁢ e 2 ⁢ i ⁢ θ ⁢ β 2 , γ ⁢ e i ⁢ θ ⁢ t ⁢ q ⁢ e - i ⁢ θ ⁢ t = ∑ n = 0 ∞ ( γ ; q ) n ( β 2 ; q ) n ⁢ C n ⁡ ( x ; β | q ) ⁢ t n q-Pochhammer-symbol 𝛾 imaginary-unit 𝜃 𝑡 𝑞 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 𝑞 \qHyperrphis 32 @ @ 𝛾 𝛽 𝛽 2 imaginary-unit 𝜃 superscript 𝛽 2 𝛾 imaginary-unit 𝜃 𝑡 𝑞 imaginary-unit 𝜃 𝑡 superscript subscript 𝑛 0 q-Pochhammer-symbol 𝛾 𝑞 𝑛 q-Pochhammer-symbol superscript 𝛽 2 𝑞 𝑛 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 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,\beta,\beta{\mathrm{e}^{2% \mathrm{i}\theta}}}{\beta^{2},\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(\beta^{2};q\right)_{n}}C_{n}\!\left(x;\beta\,|\,q\right)t^{% n}}}} {\displaystyle \frac{\qPochhammer{\gamma\expe^{\iunit\theta}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ \qHyperrphis{3}{2}@@{\gamma,\beta,\beta\expe^{2\iunit\theta}}{\beta^2,\gamma\expe^{\iunit\theta}t}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{\gamma}{q}{n}}{\qPochhammer{\beta^2}{q}{n}}\ctsqUltra{n}@{x}{\beta}{q}t^n }

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


Limit relations

Askey-Wilson polynomial to Continuous q-ultraspherical / Rogers polynomial

( β 2 ; q ) n ⁢ p n ⁡ ( x ; β 1 2 , β 1 2 ⁢ q 1 2 , - β 1 2 , - β 1 2 ⁢ q 1 2 | q ) ( β ⁢ q 1 2 , - β , - β ⁢ q 1 2 , q ; q ) n = C n ⁡ ( x ; β | q ) q-Pochhammer-symbol superscript 𝛽 2 𝑞 𝑛 Askey-Wilson-polynomial-p 𝑛 𝑥 superscript 𝛽 1 2 superscript 𝛽 1 2 superscript 𝑞 1 2 superscript 𝛽 1 2 superscript 𝛽 1 2 superscript 𝑞 1 2 𝑞 q-Pochhammer-symbol 𝛽 superscript 𝑞 1 2 𝛽 𝛽 superscript 𝑞 1 2 𝑞 𝑞 𝑛 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{\left(\beta^{2};q\right)_{n}p_% {n}\!\left(x;\beta^{\frac{1}{2}},\beta^{\frac{1}{2}}q^{\frac{1}{2}},-\beta^{% \frac{1}{2}},-\beta^{\frac{1}{2}}q^{\frac{1}{2}}\,|\,q\right)}{\left(\beta q^{% \frac{1}{2}},-\beta,-\beta q^{\frac{1}{2}},q;q\right)_{n}}=C_{n}\!\left(x;% \beta\,|\,q\right)}}} {\displaystyle \frac{\qPochhammer{\beta^2}{q}{n}\AskeyWilson{n}@{x}{\beta^{\frac{1}{2}}}{\beta^{\frac{1}{2}}q^{\frac{1}{2}}}{ -\beta^{\frac{1}{2}}}{-\beta^{\frac{1}{2}}q^{\frac{1}{2}}}{q}} {\qPochhammer{\beta q^{\frac{1}{2}},-\beta,-\beta q^{\frac{1}{2}},q}{q}{n}}=\ctsqUltra{n}@{x}{\beta}{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} }

Continuous q-ultraspherical / Rogers polynomial to Continuous q-Hermite polynomial

lim β → 0 ⁡ C n ⁡ ( x ; β | q ) = H n ⁡ ( x | q ) ( q ; q ) n subscript → 𝛽 0 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 continuous-q-Hermite-polynomial-H 𝑛 𝑥 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\lim_{\beta\rightarrow 0}C_{n}\!% \left(x;\beta\,|\,q\right)=\frac{H_{n}\!\left(x\,|\,q\right)}{\left(q;q\right)% _{n}}}}} {\displaystyle \lim_{\beta\rightarrow 0}\ctsqUltra{n}@{x}{\beta}{q}=\frac{\ctsqHermite{n}@{x}{q}}{\qPochhammer{q}{q}{n}} }

Continuous q-ultraspherical / Rogers polynomial to Gegenbauer /

lim q → 1 ⁡ C n ⁡ ( x ; q λ | q ) = C n λ ⁡ ( x ) subscript → 𝑞 1 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 superscript 𝑞 𝜆 𝑞 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\par\lim_{q\rightarrow 1}C_{n}\!% \left(x;q^{\lambda}\,|\,q\right)=C^{\lambda}_{n}\left(x\right)}}} {\displaystyle \lim_{q\rightarrow 1}\ctsqUltra{n}@{x}{q^{\lambda}}{q}=\Ultra{\lambda}{n}@{x} }

Remarks

C n ⁡ ( x ; β | q ) = ∑ k = 0 n ( β ; q ) k ⁢ ( β ; q ) n - k ( q ; q ) k ⁢ ( q ; q ) n - k ⁢ e i ⁢ ( n - 2 ⁢ k ) ⁢ θ continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 superscript subscript 𝑘 0 𝑛 q-Pochhammer-symbol 𝛽 𝑞 𝑘 q-Pochhammer-symbol 𝛽 𝑞 𝑛 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑛 𝑘 imaginary-unit 𝑛 2 𝑘 𝜃 {\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;\beta\,|\,q\right)=% \sum_{k=0}^{n}\frac{\left(\beta;q\right)_{k}\left(\beta;q\right)_{n-k}}{\left(% q;q\right)_{k}\left(q;q\right)_{n-k}}{\mathrm{e}^{\mathrm{i}(n-2k)\theta}}}}} {\displaystyle \ctsqUltra{n}@{x}{\beta}{q}=\sum_{k=0}^n\frac{\qPochhammer{\beta}{q}{k}\qPochhammer{\beta}{q}{n-k}}{\qPochhammer{q}{q}{k}\qPochhammer{q}{q}{n-k}} \expe^{\iunit(n-2k)\theta} }

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


P n ( α , α ) ⁡ ( x | q ) ⟶ q α + 1 2 → β ( β ⁢ q 1 2 ; q ) n ( β 2 ; q ) n ⁢ β 1 2 ⁢ n ⁢ C n ⁡ ( x ; β | q ) superscript ⟶ → superscript 𝑞 𝛼 1 2 𝛽 continuous-q-Jacobi-polynomial-P 𝛼 𝛼 𝑛 𝑥 𝑞 q-Pochhammer-symbol 𝛽 superscript 𝑞 1 2 𝑞 𝑛 q-Pochhammer-symbol superscript 𝛽 2 𝑞 𝑛 superscript 𝛽 1 2 𝑛 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\alpha)}_{n}\!\left(x|q% \right)\lx@stackrel{{\scriptstyle q^{\alpha+\frac{1}{2}}\rightarrow\beta}}{{% \longrightarrow}}\frac{\left(\beta q^{\frac{1}{2}};q\right)_{n}}{\left(\beta^{% 2};q\right)_{n}}\beta^{\frac{1}{2}n}C_{n}\!\left(x;\beta\,|\,q\right)}}} {\displaystyle \ctsqJacobi{\alpha}{\alpha}{n}@{x}{q}\stackrel{q^{\alpha+\frac{1}{2}}\rightarrow\beta}{\longrightarrow} \frac{\qPochhammer{\beta q^{\frac{1}{2}}}{q}{n}}{\qPochhammer{\beta^2}{q}{n}}\beta^{\frac{1}{2}n}\ctsqUltra{n}@{x}{\beta}{q} }
C n ⁡ ( x ; q α + 1 2 | q ) = ( q 2 ⁢ α + 1 ; q ) n ( q α + 1 ; q ) n ⁢ q ( 1 2 ⁢ α + 1 4 ) ⁢ n ⁢ P n ( α , α ) ⁡ ( x | q ) continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 superscript 𝑞 𝛼 1 2 𝑞 q-Pochhammer-symbol superscript 𝑞 2 𝛼 1 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 superscript 𝑞 1 2 𝛼 1 4 𝑛 continuous-q-Jacobi-polynomial-P 𝛼 𝛼 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;q^{\alpha+\frac{1}{2% }}\,|\,q\right)=\frac{\left(q^{2\alpha+1};q\right)_{n}}{\left(q^{\alpha+1};q% \right)_{n}q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}P^{(\alpha,\alpha)}_{n}\!\left% (x|q\right)}}} {\displaystyle \ctsqUltra{n}@{x}{q^{\alpha+\frac{1}{2}}}{q}= \frac{\qPochhammer{q^{2\alpha+1}}{q}{n}}{\qPochhammer{q^{\alpha+1}}{q}{n}q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}\ctsqJacobi{\alpha}{\alpha}{n}@{x}{q} }
C n ⁡ ( x ; β | q - 1 ) = ( β ⁢ q ) n ⁢ C n ⁡ ( x ; β - 1 | q ) continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 superscript 𝑞 1 superscript 𝛽 𝑞 𝑛 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 superscript 𝛽 1 𝑞 {\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;\beta\,|\,q^{-1}% \right)=(\beta q)^{n}C_{n}\!\left(x;\beta^{-1}\,|\,q\right)}}} {\displaystyle \ctsqUltra{n}@{x}{\beta}{q^{-1}}=(\beta q)^n\ctsqUltra{n}@{x}{\beta^{-1}}{q} }
C n ⁡ ( x ; q | q ) = sin ⁡ ( n + 1 ) ⁢ θ sin ⁡ θ = U n ⁡ ( x ) continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝑞 𝑞 𝑛 1 𝜃 𝜃 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;q\,|\,q\right)=\frac% {\sin\left(n+1\right)\theta}{\sin\theta}=U_{n}\left(x\right)}}} {\displaystyle \ctsqUltra{n}@{x}{q}{q}=\frac{\sin@{n+1}\theta}{\sin@@{\theta}}=\ChebyU{n}@{x} }

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


lim β → 1 ⁡ 1 - q n 2 ⁢ ( 1 - β ) ⁢ C n ⁡ ( x ; β | q ) = cos ⁡ n ⁢ θ = T n ⁡ ( x ) , x = cos ⁡ θ formulae-sequence subscript → 𝛽 1 1 superscript 𝑞 𝑛 2 1 𝛽 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 𝑛 𝜃 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle\lim_{\beta\rightarrow 1}\frac{1-q^{% n}}{2(1-\beta)}C_{n}\!\left(x;\beta\,|\,q\right)=\cos n\theta=T_{n}\left(x% \right),\quad x=\cos\theta}}} {\displaystyle \lim_{\beta\rightarrow 1}\frac{1-q^n}{2(1-\beta)}\ctsqUltra{n}@{x}{\beta}{q}=\cos@@{n}\theta =\ChebyT{n}@{x},\quad x=\cos@@{\theta} }

Constraint(s): n = 1 , 2 , 3 , … 𝑛 1 2 3 … {\displaystyle{\displaystyle{\displaystyle n=1,2,3,\ldots}}}


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} }
C n ⁡ ( x ; q 1 2 | q ) = q - 1 4 ⁢ n ⁢ P n ⁡ ( x | q ) continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 superscript 𝑞 1 2 𝑞 superscript 𝑞 1 4 𝑛 continuous-q-Legendre-polynomial-P 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;q^{\frac{1}{2}}\,|\,% q\right)=q^{-\frac{1}{4}n}P_{n}\!\left(x|q\right)}}} {\displaystyle \ctsqUltra{n}@{x}{q^{\frac{1}{2}}}{q}=q^{-\frac{1}{4}n}\ctsqLegendre{n}@{x}{q} }

Koornwinder Addendum: Continuous q-ultraspherical / Rogers

Re: (14.10.17)

C n ⁡ ( cos ⁡ θ ; β | q ) = ( β 2 ; q ) n ( q ; q ) n ⁢ β - 1 2 ⁢ n ⁢ \qHyperrphis ⁢ 43 ⁢ @ ⁢ @ ⁢ q - 1 2 ⁢ n , β ⁢ q 1 2 ⁢ n , β 1 2 ⁢ e i ⁢ θ , β 1 2 ⁢ e - i ⁢ θ - β , β 1 2 ⁢ q 1 4 , - β 1 2 ⁢ q 1 4 ⁢ q 1 2 ⁢ q 1 2 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝜃 𝛽 𝑞 q-Pochhammer-symbol superscript 𝛽 2 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝛽 1 2 𝑛 \qHyperrphis 43 @ @ superscript 𝑞 1 2 𝑛 𝛽 superscript 𝑞 1 2 𝑛 superscript 𝛽 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 imaginary-unit 𝜃 𝛽 superscript 𝛽 1 2 superscript 𝑞 1 4 superscript 𝛽 1 2 superscript 𝑞 1 4 superscript 𝑞 1 2 superscript 𝑞 1 2 {\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(\cos\theta;\beta\,|\,q% \right)=\frac{\left(\beta^{2};q\right)_{n}}{\left(q;q\right)_{n}}\beta^{-\frac% {1}{2}n}\qHyperrphis{4}{3}@@{q^{-\frac{1}{2}n},\beta q^{\frac{1}{2}n},\beta^{% \frac{1}{2}}{\mathrm{e}^{\mathrm{i}\theta}},\beta^{\frac{1}{2}}{\mathrm{e}^{-% \mathrm{i}\theta}}}{-\beta,\beta^{\frac{1}{2}}q^{\frac{1}{4}},-\beta^{\frac{1}% {2}}q^{\frac{1}{4}}}{q^{\frac{1}{2}}}{q^{\frac{1}{2}}}}}} {\displaystyle \ctsqUltra{n}@{\cos@@{\theta}}{\beta }{ q}= \frac{\qPochhammer{\beta^2}{q}{n}}{\qPochhammer{q}{q}{n}} \beta^{-\frac12 n} \qHyperrphis{4}{3}@@{q^{-\frac12 n},\beta q^{\frac12 n},\beta^\frac12 \expe^{\iunit\theta},\beta^\frac12 \expe^{-\iunit\theta}}{-\beta,\beta^\frac12 q^{\frac14},-\beta^\frac12 q^{\frac14}}{q^\frac12}{q^\frac12} }

Continuous q-Jacobi: Special cases: Special value

C n ⁡ ( 1 2 ⁢ ( β 1 2 + β - 1 2 ) ; β | q ) = ( β 2 ; q ) n ( q ; q ) n ⁢ β - 1 2 ⁢ n continuous-q-ultraspherical-Rogers-polynomial 𝑛 1 2 superscript 𝛽 1 2 superscript 𝛽 1 2 𝛽 𝑞 q-Pochhammer-symbol superscript 𝛽 2 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝛽 1 2 𝑛 {\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(\frac{1}{2}(\beta^{% \frac{1}{2}}+\beta^{-\frac{1}{2}});\beta\,|\,q\right)=\frac{\left(\beta^{2};q% \right)_{n}}{\left(q;q\right)_{n}}\beta^{-\frac{1}{2}n}}}} {\displaystyle \ctsqUltra{n}@{\frac12(\beta^\frac12+\beta^{-\frac12})}{\beta }{ q} =\frac{\qPochhammer{\beta^2}{q}{n}}{\qPochhammer{q}{q}{n}} \beta^{-\frac12 n} }

Re: (14.10.21)

1 - β ⁢ z 2 1 - z 2 ⁢ C n ⁡ [ q 1 2 ⁢ z ; β | q ] + 1 - β ⁢ z - 2 1 - z - 2 ⁢ C n ⁡ [ q - 1 2 ⁢ z ; β | q ] = ( q - 1 2 ⁢ n + q 1 2 ⁢ n ⁢ β ) ⁢ C n ⁡ [ z ; β | q ] 1 𝛽 superscript 𝑧 2 1 superscript 𝑧 2 continuous-q-ultraspherical-Rogers-polynomial-exponential-argument 𝑛 superscript 𝑞 1 2 𝑧 𝛽 𝑞 1 𝛽 superscript 𝑧 2 1 superscript 𝑧 2 continuous-q-ultraspherical-Rogers-polynomial-exponential-argument 𝑛 superscript 𝑞 1 2 𝑧 𝛽 𝑞 superscript 𝑞 1 2 𝑛 superscript 𝑞 1 2 𝑛 𝛽 continuous-q-ultraspherical-Rogers-polynomial-exponential-argument 𝑛 𝑧 𝛽 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{1-\beta z^{2}}{1-z^{2}}C_{n}\!% \left[q^{\frac{1}{2}}z;\beta\,|\,q\right]+\frac{1-\beta z^{-2}}{1-z^{-2}}C_{n}% \!\left[q^{-\frac{1}{2}}z;\beta\,|\,q\right]=(q^{-\frac{1}{2}n}+q^{\frac{1}{2}% n}\beta)C_{n}\!\left[z;\beta\,|\,q\right]}}} {\displaystyle \frac{1-\beta z^2}{1-z^2} \ctsqUltrae{n}@{q^\frac12 z}{\beta }{ q}+ \frac{1-\beta z^{-2}}{1-z^{-2}} \ctsqUltrae{n}@{q^{-\frac12}z}{\beta }{ q}= (q^{-\frac12 n}+q^{\frac12 n} \beta) \ctsqUltrae{n}@{z}{\beta }{ q} }

Re: (14.10.23)

C n ⁡ [ q 1 2 ⁢ z ; β | q ] - C n ⁡ [ q - 1 2 ⁢ z ; β | q ] = q - 1 2 ⁢ n ⁢ ( β - 1 ) ⁢ ( z - z - 1 ) ⁢ C n - 1 ⁡ [ z ; q ⁢ β | q ] continuous-q-ultraspherical-Rogers-polynomial-exponential-argument 𝑛 superscript 𝑞 1 2 𝑧 𝛽 𝑞 continuous-q-ultraspherical-Rogers-polynomial-exponential-argument 𝑛 superscript 𝑞 1 2 𝑧 𝛽 𝑞 superscript 𝑞 1 2 𝑛 𝛽 1 𝑧 superscript 𝑧 1 continuous-q-ultraspherical-Rogers-polynomial-exponential-argument 𝑛 1 𝑧 𝑞 𝛽 𝑞 {\displaystyle{\displaystyle{\displaystyle C_{n}\!\left[q^{\frac{1}{2}}z;\beta% \,|\,q\right]-C_{n}\!\left[q^{-\frac{1}{2}}z;\beta\,|\,q\right]=q^{-\frac{1}{2% }n}(\beta-1)(z-z^{-1})C_{n-1}\!\left[z;q\beta\,|\,q\right]}}} {\displaystyle \ctsqUltrae{n}@{q^\frac12 z}{\beta }{ q}-\ctsqUltrae{n}@{q^{-\frac12}z}{\beta }{ q}= q^{-\frac12 n}(\beta-1)(z-z^{-1})\ctsqUltrae{n-1}@{z}{q\beta }{ q} }
( β + 1 ) ⁢ C n ⁡ [ q 1 2 ⁢ z ; β | q ] = ( q - 1 2 ⁢ n + q 1 2 ⁢ n ⁢ β ) ⁢ C n ⁡ [ z ; β | q ] + q - 1 2 ⁢ n ⁢ ( β - 1 ) ⁢ ( z - β ⁢ z - 1 ) ⁢ C n - 1 ⁡ [ z ; q ⁢ β | q ] 𝛽 1 continuous-q-ultraspherical-Rogers-polynomial-exponential-argument 𝑛 superscript 𝑞 1 2 𝑧 𝛽 𝑞 superscript 𝑞 1 2 𝑛 superscript 𝑞 1 2 𝑛 𝛽 continuous-q-ultraspherical-Rogers-polynomial-exponential-argument 𝑛 𝑧 𝛽 𝑞 superscript 𝑞 1 2 𝑛 𝛽 1 𝑧 𝛽 superscript 𝑧 1 continuous-q-ultraspherical-Rogers-polynomial-exponential-argument 𝑛 1 𝑧 𝑞 𝛽 𝑞 {\displaystyle{\displaystyle{\displaystyle(\beta+1)C_{n}\!\left[q^{\frac{1}{2}% }z;\beta\,|\,q\right]=(q^{-\frac{1}{2}n}+q^{\frac{1}{2}n}\beta)C_{n}\!\left[z;% \beta\,|\,q\right]+q^{-\frac{1}{2}n}(\beta-1)(z-\beta z^{-1})C_{n-1}\!\left[z;% q\beta\,|\,q\right]}}} {\displaystyle (\beta+1)\ctsqUltrae{n}@{q^\frac12 z}{\beta }{ q}=(q^{-\frac12 n}+q^{\frac12 n}\beta)\ctsqUltrae{n}@{z}{\beta }{ q} +q^{-\frac12 n}(\beta-1)(z-\beta z^{-1})\ctsqUltrae{n-1}@{z}{q\beta }{ q} }
( β + 1 ) ⁢ C n ⁡ [ q - 1 2 ⁢ z ; β | q ] = ( q - 1 2 ⁢ n + q 1 2 ⁢ n ⁢ β ) ⁢ C n ⁡ [ z ; β | q ] + q - 1 2 ⁢ n ⁢ ( β - 1 ) ⁢ ( z - 1 - β ⁢ z ) ⁢ C n - 1 ⁡ [ z ; q ⁢ β | q ] 𝛽 1 continuous-q-ultraspherical-Rogers-polynomial-exponential-argument 𝑛 superscript 𝑞 1 2 𝑧 𝛽 𝑞 superscript 𝑞 1 2 𝑛 superscript 𝑞 1 2 𝑛 𝛽 continuous-q-ultraspherical-Rogers-polynomial-exponential-argument 𝑛 𝑧 𝛽 𝑞 superscript 𝑞 1 2 𝑛 𝛽 1 superscript 𝑧 1 𝛽 𝑧 continuous-q-ultraspherical-Rogers-polynomial-exponential-argument 𝑛 1 𝑧 𝑞 𝛽 𝑞 {\displaystyle{\displaystyle{\displaystyle(\beta+1)C_{n}\!\left[q^{-\frac{1}{2% }}z;\beta\,|\,q\right]=(q^{-\frac{1}{2}n}+q^{\frac{1}{2}n}\beta)C_{n}\!\left[z% ;\beta\,|\,q\right]+q^{-\frac{1}{2}n}(\beta-1)(z^{-1}-\beta z)C_{n-1}\!\left[z% ;q\beta\,|\,q\right]}}} {\displaystyle (\beta+1)\ctsqUltrae{n}@{q^{-\frac12}z}{\beta }{ q}=(q^{-\frac12 n}+q^{\frac12 n}\beta)\ctsqUltrae{n}@{z}{\beta }{ q} +q^{-\frac12 n}(\beta-1)(z^{-1}-\beta z)\ctsqUltrae{n-1}@{z}{q\beta }{ q} }

Trigonometric representation

C n ⁡ ( cos ⁡ θ ; β | q ) = ∑ k = 0 n ( β ; q ) k ⁢ ( β ; q ) n - k ( q ; q ) k ⁢ ( q ; q ) n - k ⁢ e i ⁢ ( n - 2 ⁢ k ) ⁢ θ continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝜃 𝛽 𝑞 superscript subscript 𝑘 0 𝑛 q-Pochhammer-symbol 𝛽 𝑞 𝑘 q-Pochhammer-symbol 𝛽 𝑞 𝑛 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑛 𝑘 imaginary-unit 𝑛 2 𝑘 𝜃 {\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(\cos\theta;\beta\,|\,q% \right)=\sum_{k=0}^{n}\frac{\left(\beta;q\right)_{k}\left(\beta;q\right)_{n-k}% }{\left(q;q\right)_{k}\left(q;q\right)_{n-k}}{\mathrm{e}^{\mathrm{i}(n-2k)% \theta}}}}} {\displaystyle \ctsqUltra{n}@{\cos@@{\theta}}{\beta }{ q}=\sum_{k=0}^n \frac{\qPochhammer{\beta}{q}{k} \qPochhammer{\beta}{q}{n-k}}{\qPochhammer{q}{q}{k} \qPochhammer{q}{q}{n-k}} \expe^{\iunit(n-2k)\theta} }

Limit for q\downarrow-1

lim q ↑ 1 ⁡ C 2 ⁢ m ⁡ ( x ; - q \la | - q ) = C m 1 2 ⁢ ( \la + 1 ) ⁡ ( 2 ⁢ x 2 - 1 ) + C m - 1 1 2 ⁢ ( \la + 1 ) ⁡ ( 2 ⁢ x 2 - 1 ) subscript ↑ 𝑞 1 continuous-q-ultraspherical-Rogers-polynomial 2 𝑚 𝑥 superscript 𝑞 \la 𝑞 ultraspherical-Gegenbauer-polynomial 1 2 \la 1 𝑚 2 superscript 𝑥 2 1 ultraspherical-Gegenbauer-polynomial 1 2 \la 1 𝑚 1 2 superscript 𝑥 2 1 {\displaystyle{\displaystyle{\displaystyle\lim_{q\uparrow 1}C_{2m}\!\left(x;-q% ^{\la}\,|\,-q\right)=C^{\frac{1}{2}(\la+1)}_{m}\left(2x^{2}-1\right)+C^{\frac{% 1}{2}(\la+1)}_{m-1}\left(2x^{2}-1\right)}}} {\displaystyle \lim_{q\uparrow1} \ctsqUltra{2m}@{x}{-q^\la }{ -q}= \Ultra{\half(\la+1)}{m}@{2x^2-1}+\Ultra{\half(\la+1)}{m-1}@{2x^2-1} }
lim q ↑ 1 ⁡ C 2 ⁢ m + 1 ⁡ ( x ; - q \la | - q ) = 2 ⁢ x ⁢ C m 1 2 ⁢ ( \la + 1 ) ⁡ ( 2 ⁢ x 2 - 1 ) subscript ↑ 𝑞 1 continuous-q-ultraspherical-Rogers-polynomial 2 𝑚 1 𝑥 superscript 𝑞 \la 𝑞 2 𝑥 ultraspherical-Gegenbauer-polynomial 1 2 \la 1 𝑚 2 superscript 𝑥 2 1 {\displaystyle{\displaystyle{\displaystyle\lim_{q\uparrow 1}C_{2m+1}\!\left(x;% -q^{\la}\,|\,-q\right)=2xC^{\frac{1}{2}(\la+1)}_{m}\left(2x^{2}-1\right)}}} {\displaystyle \lim_{q\uparrow1} \ctsqUltra{2m+1}@{x}{-q^\la }{ -q}= 2x \Ultra{\half(\la+1)}{m}@{2x^2-1} }
lim q ↑ 1 ⁡ C 2 ⁢ m ⁡ ( x ; - q λ | - q ) = ( λ ) m ( 1 2 ⁢ λ ) m ⁢ P m ( 1 2 ⁢ λ , 1 2 ⁢ λ - 1 ) ⁡ ( 2 ⁢ x 2 - 1 ) subscript ↑ 𝑞 1 continuous-q-ultraspherical-Rogers-polynomial 2 𝑚 𝑥 superscript 𝑞 𝜆 𝑞 Pochhammer-symbol 𝜆 𝑚 Pochhammer-symbol 1 2 𝜆 𝑚 Jacobi-polynomial-P 1 2 𝜆 1 2 𝜆 1 𝑚 2 superscript 𝑥 2 1 {\displaystyle{\displaystyle{\displaystyle\lim_{q\uparrow 1}C_{2m}\!\left(x;-q% ^{\lambda}\,|\,-q\right)=\frac{{\left(\lambda\right)_{m}}}{{\left(\frac{1}{2}% \lambda\right)_{m}}}P^{(\frac{1}{2}\lambda,\frac{1}{2}\lambda-1)}_{m}\left(2x^% {2}-1\right)}}} {\displaystyle \lim_{q\uparrow1} \ctsqUltra{2m}@{x}{-q^\lambda }{ -q}= \frac{\pochhammer{\lambda}{m}}{\pochhammer{\frac12\lambda}{m}} \Jacobi{\frac12\lambda}{\frac12\lambda-1}{m}@{2x^2-1} }
lim q ↑ 1 ⁡ C 2 ⁢ m + 1 ⁡ ( x ; - q λ | - q ) = 2 ⁢ ( λ + 1 ) m ( 1 2 ⁢ λ + 1 ) m ⁢ x ⁢ P m ( 1 2 ⁢ λ , 1 2 ⁢ λ ) ⁡ ( 2 ⁢ x 2 - 1 ) subscript ↑ 𝑞 1 continuous-q-ultraspherical-Rogers-polynomial 2 𝑚 1 𝑥 superscript 𝑞 𝜆 𝑞 2 Pochhammer-symbol 𝜆 1 𝑚 Pochhammer-symbol 1 2 𝜆 1 𝑚 𝑥 Jacobi-polynomial-P 1 2 𝜆 1 2 𝜆 𝑚 2 superscript 𝑥 2 1 {\displaystyle{\displaystyle{\displaystyle\lim_{q\uparrow 1}C_{2m+1}\!\left(x;% -q^{\lambda}\,|\,-q\right)=2\frac{{\left(\lambda+1\right)_{m}}}{{\left(\frac{1% }{2}\lambda+1\right)_{m}}}xP^{(\frac{1}{2}\lambda,\frac{1}{2}\lambda)}_{m}% \left(2x^{2}-1\right)}}} {\displaystyle \lim_{q\uparrow1} \ctsqUltra{2m+1}@{x}{-q^\lambda }{ -q}= 2 \frac{\pochhammer{\lambda+1}{m}}{\pochhammer{\frac12\lambda+1}{m}} x \Jacobi{\frac12\lambda}{\frac12\lambda}{m}@{2x^2-1} }
lim q ↑ 1 ⁡ C n ⁡ ( x ; - q λ | - q ) = c ⁢ o ⁢ n ⁢ s ⁢ t ⁢ S n ( 1 2 ⁢ λ , 1 2 ⁢ λ - 1 ) ⁡ ( x ) subscript ↑ 𝑞 1 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 superscript 𝑞 𝜆 𝑞 c 𝑜 𝑛 𝑠 𝑡 generalized-Gegenbauer-polynomial-S 1 2 𝜆 1 2 𝜆 1 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\lim_{q\uparrow 1}C_{n}\!\left(x;-q^% {\lambda}\,|\,-q\right)=\mathrm{c}onstS^{(\frac{1}{2}\lambda,\frac{1}{2}% \lambda-1)}_{n}\left(x\right)}}} {\displaystyle \lim_{q\uparrow1} \ctsqUltra{n}@{x}{-q^\lambda }{ -q} =\mathrm const \GenGegenbauer{\frac12\lambda}{\frac12\lambda-1}{n}@{x} }

Continuous q-Legendre

Basic hypergeometric representation

P n ⁡ ( x | q ) = \qHyperrphis ⁢ 43 ⁢ @ ⁢ @ ⁢ q - n , q n + 1 , q 1 4 ⁢ e i ⁢ θ ⁢ q 1 4 ⁢ e - i ⁢ θ ⁢ q , - q 1 2 , - q ⁢ q ⁢ q continuous-q-Legendre-polynomial-P 𝑛 𝑥 𝑞 \qHyperrphis 43 @ @ superscript 𝑞 𝑛 superscript 𝑞 𝑛 1 superscript 𝑞 1 4 imaginary-unit 𝜃 superscript 𝑞 1 4 imaginary-unit 𝜃 𝑞 superscript 𝑞 1 2 𝑞 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x|q\right)=% \qHyperrphis{4}{3}@@{q^{-n},q^{n+1},q^{\frac{1}{4}}{\mathrm{e}^{\mathrm{i}% \theta}}q^{\frac{1}{4}}{\mathrm{e}^{-\mathrm{i}\theta}}}{q,-q^{\frac{1}{2}},-q% }{q}{q}}}} {\displaystyle \ctsqLegendre{n}@{x}{q}=\qHyperrphis{4}{3}@@{q^{-n},q^{n+1},q^{\frac{1}{4}}\expe^{\iunit\theta} q^{\frac{1}{4}}\expe^{-\iunit\theta}}{q,-q^{\frac{1}{2}},-q}{q}{q} }

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


Orthogonality relation(s)

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

Substitution(s): w ⁢ ( x ; a | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ q 1 4 ⁢ e i ⁢ θ ⁢ a ⁢ q 3 4 ⁢ e i ⁢ θ , - a ⁢ q 1 4 ⁢ e i ⁢ θ - a ⁢ q 3 4 ⁢ e i ⁢ θ ; q ) ∞ | 2 = | ( e i ⁢ θ , - e i ⁢ θ ; q 1 2 ) ∞ ( a ⁢ q 1 4 ⁢ e i ⁢ θ , - a ⁢ q 1 4 ⁢ e i ⁢ θ ; q 1 2 ) ∞ | 2 = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a 2 ⁢ q 1 2 ⁢ e 2 ⁢ i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ⁢ q 1 4 ) ⁢ h ⁢ ( x , a ⁢ q 3 4 ) ⁢ h ⁢ ( x , - a ⁢ q 1 4 ) ⁢ h ⁢ ( x , - a ⁢ q 3 4 ) 𝑤 𝑥 conditional 𝑎 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 superscript 𝑞 1 4 imaginary-unit 𝜃 𝑎 superscript 𝑞 3 4 imaginary-unit 𝜃 𝑎 superscript 𝑞 1 4 imaginary-unit 𝜃 𝑎 superscript 𝑞 3 4 imaginary-unit 𝜃 𝑞 2 superscript q-Pochhammer-symbol imaginary-unit 𝜃 imaginary-unit 𝜃 superscript 𝑞 1 2 q-Pochhammer-symbol 𝑎 superscript 𝑞 1 4 imaginary-unit 𝜃 𝑎 superscript 𝑞 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 2 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol superscript 𝑎 2 superscript 𝑞 1 2 2 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 superscript 𝑞 1 4 ℎ 𝑥 𝑎 superscript 𝑞 3 4 ℎ 𝑥 𝑎 superscript 𝑞 1 4 ℎ 𝑥 𝑎 superscript 𝑞 3 4 {\displaystyle{\displaystyle{\displaystyle w(x;a|q)=\left|\frac{\left({\mathrm% {e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(aq^{\frac{1}{4}}{\mathrm{e}^% {\mathrm{i}\theta}}aq^{\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}},-aq^{\frac{% 1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}-aq^{\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(aq^{\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},-aq^{\frac{1}{4}}{% \mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\right)_{\infty}}\right|^{2}=% \left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a^% {2}q^{\frac{1}{2}}{\mathrm{e}^{2\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,aq^{\frac% {1}{4}})h(x,aq^{\frac{3}{4}})h(x,-aq^{\frac{1}{4}})h(x,-aq^{\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 ⁢ ( 1 - q n + 1 2 ) ⁢ x ⁢ P n ⁡ ( x | q ) = q - 1 4 ⁢ ( 1 - q n + 1 ) ⁢ P n + 1 ⁡ ( x | q ) + q 1 4 ⁢ ( 1 - q n ) ⁢ P n - 1 ⁡ ( x | q ) 2 1 superscript 𝑞 𝑛 1 2 𝑥 continuous-q-Legendre-polynomial-P 𝑛 𝑥 𝑞 superscript 𝑞 1 4 1 superscript 𝑞 𝑛 1 continuous-q-Legendre-polynomial-P 𝑛 1 𝑥 𝑞 superscript 𝑞 1 4 1 superscript 𝑞 𝑛 continuous-q-Legendre-polynomial-P 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle 2(1-q^{n+\frac{1}{2}})xP_{n}\!\left% (x|q\right)=q^{-\frac{1}{4}}(1-q^{n+1})P_{n+1}\!\left(x|q\right)+q^{\frac{1}{4% }}(1-q^{n})P_{n-1}\!\left(x|q\right)}}} {\displaystyle 2(1-q^{n+\frac{1}{2}})x\ctsqLegendre{n}@{x}{q}=q^{-\frac{1}{4}}(1-q^{n+1})\ctsqLegendre{n+1}@{x}{q} +q^{\frac{1}{4}}(1-q^n)\ctsqLegendre{n-1}@{x}{q} }

Monic recurrence relation

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

q-Difference equation

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

Substitution(s): y ⁢ ( x ) = P n ⁡ ( x | q ) 𝑦 𝑥 continuous-q-Legendre-polynomial-P 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=P_{n}\!\left(x|q\right)}}} &

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}}}}}} &
λ n = 4 ⁢ q - n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 - q n + 1 ) subscript 𝜆 𝑛 4 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\lambda_{n}=4q^{-n+1}(1-q^{n})(1-q^{% n+1})}}} &
w ⁢ ( x ; a | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ q 1 4 ⁢ e i ⁢ θ ⁢ a ⁢ q 3 4 ⁢ e i ⁢ θ , - a ⁢ q 1 4 ⁢ e i ⁢ θ - a ⁢ q 3 4 ⁢ e i ⁢ θ ; q ) ∞ | 2 = | ( e i ⁢ θ , - e i ⁢ θ ; q 1 2 ) ∞ ( a ⁢ q 1 4 ⁢ e i ⁢ θ , - a ⁢ q 1 4 ⁢ e i ⁢ θ ; q 1 2 ) ∞ | 2 = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a 2 ⁢ q 1 2 ⁢ e 2 ⁢ i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ⁢ q 1 4 ) ⁢ h ⁢ ( x , a ⁢ q 3 4 ) ⁢ h ⁢ ( x , - a ⁢ q 1 4 ) ⁢ h ⁢ ( x , - a ⁢ q 3 4 ) 𝑤 𝑥 conditional 𝑎 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 superscript 𝑞 1 4 imaginary-unit 𝜃 𝑎 superscript 𝑞 3 4 imaginary-unit 𝜃 𝑎 superscript 𝑞 1 4 imaginary-unit 𝜃 𝑎 superscript 𝑞 3 4 imaginary-unit 𝜃 𝑞 2 superscript q-Pochhammer-symbol imaginary-unit 𝜃 imaginary-unit 𝜃 superscript 𝑞 1 2 q-Pochhammer-symbol 𝑎 superscript 𝑞 1 4 imaginary-unit 𝜃 𝑎 superscript 𝑞 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 2 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol superscript 𝑎 2 superscript 𝑞 1 2 2 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 superscript 𝑞 1 4 ℎ 𝑥 𝑎 superscript 𝑞 3 4 ℎ 𝑥 𝑎 superscript 𝑞 1 4 ℎ 𝑥 𝑎 superscript 𝑞 3 4 {\displaystyle{\displaystyle{\displaystyle w(x;a|q)=\left|\frac{\left({\mathrm% {e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(aq^{\frac{1}{4}}{\mathrm{e}^% {\mathrm{i}\theta}}aq^{\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}},-aq^{\frac{% 1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}-aq^{\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(aq^{\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},-aq^{\frac{1}{4}}{% \mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\right)_{\infty}}\right|^{2}=% \left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a^% {2}q^{\frac{1}{2}}{\mathrm{e}^{2\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,aq^{\frac% {1}{4}})h(x,aq^{\frac{3}{4}})h(x,-aq^{\frac{1}{4}})h(x,-aq^{\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 ; 1 | q ) ⁢ P n ⁡ ( x | q ) = ( q - 1 2 ) n ⁢ q 1 4 ⁢ n 2 ( q , - q 1 2 , - q ; q ) n ⁢ ( D q ) n ⁢ [ w ~ ⁢ ( x ; q 1 2 ⁢ n | q ) ] ~ 𝑤 𝑥 conditional 1 𝑞 continuous-q-Legendre-polynomial-P 𝑛 𝑥 𝑞 superscript 𝑞 1 2 𝑛 superscript 𝑞 1 4 superscript 𝑛 2 q-Pochhammer-symbol 𝑞 superscript 𝑞 1 2 𝑞 𝑞 𝑛 superscript subscript 𝐷 𝑞 𝑛 delimited-[] ~ 𝑤 𝑥 conditional superscript 𝑞 1 2 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;1|q)P_{n}\!\left(x|q% \right)=\left(\frac{q-1}{2}\right)^{n}\frac{q^{\frac{1}{4}n^{2}}}{\left(q,-q^{% \frac{1}{2}},-q;q\right)_{n}}\left(D_{q}\right)^{n}\left[{\tilde{w}}(x;q^{% \frac{1}{2}n}|q)\right]}}} {\displaystyle {\tilde w}(x;1|q)\ctsqLegendre{n}@{x}{q}=\left(\frac{q-1}{2}\right)^n \frac{q^{\frac{1}{4}n^2}}{\qPochhammer{q,-q^{\frac{1}{2}},-q}{q}{n}} \left(D_q\right)^n\left[{\tilde w}(x;q^{\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 ; q ) ∞ ( a ⁢ q 1 4 ⁢ e i ⁢ θ ⁢ a ⁢ q 3 4 ⁢ e i ⁢ θ , - a ⁢ q 1 4 ⁢ e i ⁢ θ - a ⁢ q 3 4 ⁢ e i ⁢ θ ; q ) ∞ | 2 = | ( e i ⁢ θ , - e i ⁢ θ ; q 1 2 ) ∞ ( a ⁢ q 1 4 ⁢ e i ⁢ θ , - a ⁢ q 1 4 ⁢ e i ⁢ θ ; q 1 2 ) ∞ | 2 = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a 2 ⁢ q 1 2 ⁢ e 2 ⁢ i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ⁢ q 1 4 ) ⁢ h ⁢ ( x , a ⁢ q 3 4 ) ⁢ h ⁢ ( x , - a ⁢ q 1 4 ) ⁢ h ⁢ ( x , - a ⁢ q 3 4 ) 𝑤 𝑥 conditional 𝑎 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 superscript 𝑞 1 4 imaginary-unit 𝜃 𝑎 superscript 𝑞 3 4 imaginary-unit 𝜃 𝑎 superscript 𝑞 1 4 imaginary-unit 𝜃 𝑎 superscript 𝑞 3 4 imaginary-unit 𝜃 𝑞 2 superscript q-Pochhammer-symbol imaginary-unit 𝜃 imaginary-unit 𝜃 superscript 𝑞 1 2 q-Pochhammer-symbol 𝑎 superscript 𝑞 1 4 imaginary-unit 𝜃 𝑎 superscript 𝑞 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 2 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol superscript 𝑎 2 superscript 𝑞 1 2 2 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 superscript 𝑞 1 4 ℎ 𝑥 𝑎 superscript 𝑞 3 4 ℎ 𝑥 𝑎 superscript 𝑞 1 4 ℎ 𝑥 𝑎 superscript 𝑞 3 4 {\displaystyle{\displaystyle{\displaystyle w(x;a|q)=\left|\frac{\left({\mathrm% {e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(aq^{\frac{1}{4}}{\mathrm{e}^% {\mathrm{i}\theta}}aq^{\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}},-aq^{\frac{% 1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}-aq^{\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(aq^{\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},-aq^{\frac{1}{4}}{% \mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\right)_{\infty}}\right|^{2}=% \left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a^% {2}q^{\frac{1}{2}}{\mathrm{e}^{2\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,aq^{\frac% {1}{4}})h(x,aq^{\frac{3}{4}})h(x,-aq^{\frac{1}{4}})h(x,-aq^{\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 1 2 ⁢ e i ⁢ θ ⁢ t ; q ) ∞ ( e i ⁢ θ ⁢ t ; q ) ∞ | 2 = ( q 1 2 ⁢ e i ⁢ θ ⁢ t , q 1 2 ⁢ e - i ⁢ θ ⁢ t ; q ) ∞ ( e i ⁢ θ ⁢ t , e - i ⁢ θ ⁢ t ; q ) ∞ = ∑ n = 0 ∞ P n ⁡ ( x | q ) q 1 4 ⁢ n ⁢ t n superscript q-Pochhammer-symbol superscript 𝑞 1 2 imaginary-unit 𝜃 𝑡 𝑞 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 𝑞 2 q-Pochhammer-symbol superscript 𝑞 1 2 imaginary-unit 𝜃 𝑡 superscript 𝑞 1 2 imaginary-unit 𝜃 𝑡 𝑞 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 imaginary-unit 𝜃 𝑡 𝑞 superscript subscript 𝑛 0 continuous-q-Legendre-polynomial-P 𝑛 𝑥 𝑞 superscript 𝑞 1 4 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left|\frac{\left(q^{\frac{1}{2}}{% \mathrm{e}^{\mathrm{i}\theta}}t;q\right)_{\infty}}{\left({\mathrm{e}^{\mathrm{% i}\theta}}t;q\right)_{\infty}}\right|^{2}=\frac{\left(q^{\frac{1}{2}}{\mathrm{% e}^{\mathrm{i}\theta}}t,q^{\frac{1}{2}}{\mathrm{e}^{-\mathrm{i}\theta}}t;q% \right)_{\infty}}{\left({\mathrm{e}^{\mathrm{i}\theta}}t,{\mathrm{e}^{-\mathrm% {i}\theta}}t;q\right)_{\infty}}=\sum_{n=0}^{\infty}\frac{P_{n}\!\left(x|q% \right)}{q^{\frac{1}{4}n}}t^{n}}}} {\displaystyle \left|\frac{\qPochhammer{q^{\frac{1}{2}}\expe^{\iunit\theta}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\right|^2 =\frac{\qPochhammer{q^{\frac{1}{2}}\expe^{\iunit\theta}t,q^{\frac{1}{2}}\expe^{-\iunit\theta}t}{q}{\infty}} {\qPochhammer{\expe^{\iunit\theta}t,\expe^{-\iunit\theta}t}{q}{\infty}} =\sum_{n=0}^{\infty}\frac{\ctsqLegendre{n}@{x}{q}}{q^{\frac{1}{4}n}}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 , q 1 2 ⁢ e 2 ⁢ i ⁢ θ ⁢ q ⁢ q ⁢ e - i ⁢ θ ⁢ t = ∑ n = 0 ∞ P n ⁡ ( x | q ) ( q ; q ) n ⁢ q 1 4 ⁢ n ⁢ t n 1 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 𝑞 \qHyperrphis 21 @ @ superscript 𝑞 1 2 superscript 𝑞 1 2 2 imaginary-unit 𝜃 𝑞 𝑞 imaginary-unit 𝜃 𝑡 superscript subscript 𝑛 0 continuous-q-Legendre-polynomial-P 𝑛 𝑥 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 1 4 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{\left({\mathrm{e}^{\mathrm{% i}\theta}}t;q\right)_{\infty}}\ \qHyperrphis{2}{1}@@{q^{\frac{1}{2}},q^{\frac{% 1}{2}}{\mathrm{e}^{2\mathrm{i}\theta}}}{q}{q}{{\mathrm{e}^{-\mathrm{i}\theta}}% t}=\sum_{n=0}^{\infty}\frac{P_{n}\!\left(x|q\right)}{\left(q;q\right)_{n}q^{% \frac{1}{4}n}}t^{n}}}} {\displaystyle \frac{1}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ \qHyperrphis{2}{1}@@{q^{\frac{1}{2}},q^{\frac{1}{2}}\expe^{2\iunit\theta}}{q}{q}{\expe^{-\iunit\theta}t} =\sum_{n=0}^{\infty}\frac{\ctsqLegendre{n}@{x}{q}}{\qPochhammer{q}{q}{n}q^{\frac{1}{4}n}}t^n }

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


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

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


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

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


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

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


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

Continuous q-Legendre polynomial to Legendre / Spherical polynomial

lim q → 1 ⁡ P n ⁡ ( x | q ) = \LegendrePoly ⁢ n ⁢ @ ⁢ x subscript → 𝑞 1 continuous-q-Legendre-polynomial-P 𝑛 𝑥 𝑞 \LegendrePoly 𝑛 @ 𝑥 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}P_{n}\!\left(x|% q\right)=\LegendrePoly{n}@{x}}}} {\displaystyle \lim_{q\rightarrow 1}\ctsqLegendre{n}@{x}{q}=\LegendrePoly{n}@{x} }

Remarks

P n ⁡ ( x | q ) = q 1 4 ⁢ n ⁢ ∑ k = 0 n ( q 1 2 ; q ) k ⁢ ( q 1 2 ; q ) n - k ( q ; q ) k ⁢ ( q ; q ) n - k ⁢ e i ⁢ ( n - 2 ⁢ k ) ⁢ θ continuous-q-Legendre-polynomial-P 𝑛 𝑥 𝑞 superscript 𝑞 1 4 𝑛 superscript subscript 𝑘 0 𝑛 q-Pochhammer-symbol superscript 𝑞 1 2 𝑞 𝑘 q-Pochhammer-symbol superscript 𝑞 1 2 𝑞 𝑛 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑛 𝑘 imaginary-unit 𝑛 2 𝑘 𝜃 {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x|q\right)=q^{\frac{1}% {4}n}\sum_{k=0}^{n}\frac{\left(q^{\frac{1}{2}};q\right)_{k}\left(q^{\frac{1}{2% }};q\right)_{n-k}}{\left(q;q\right)_{k}\left(q;q\right)_{n-k}}{\mathrm{e}^{% \mathrm{i}(n-2k)\theta}}}}} {\displaystyle \ctsqLegendre{n}@{x}{q}=q^{\frac{1}{4}n}\sum_{k=0}^n \frac{\qPochhammer{q^{\frac{1}{2}}}{q}{k}\qPochhammer{q^{\frac{1}{2}}}{q}{n-k}}{\qPochhammer{q}{q}{k}\qPochhammer{q}{q}{n-k}} \expe^{\iunit(n-2k)\theta} }

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


P n ⁡ ( x ; q ) = \qHyperrphis ⁢ 43 ⁢ @ ⁢ @ ⁢ q - n , q n + 1 , q 1 2 ⁢ e i ⁢ θ , q 1 2 ⁢ e - i ⁢ θ ⁢ q , - q , - q ⁢ q ⁢ q continuous-q-Legendre-Rahman-polynomial-P 𝑛 𝑥 𝑞 \qHyperrphis 43 @ @ superscript 𝑞 𝑛 superscript 𝑞 𝑛 1 superscript 𝑞 1 2 imaginary-unit 𝜃 superscript 𝑞 1 2 imaginary-unit 𝜃 𝑞 𝑞 𝑞 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;q\right)=% \qHyperrphis{4}{3}@@{q^{-n},q^{n+1},q^{\frac{1}{2}}{\mathrm{e}^{\mathrm{i}% \theta}},q^{\frac{1}{2}}{\mathrm{e}^{-\mathrm{i}\theta}}}{q,-q,-q}{q}{q}}}} {\displaystyle \ctsqLegendreRahman{n}@{x}{q}=\qHyperrphis{4}{3}@@{q^{-n},q^{n+1},q^{\frac{1}{2}}\expe^{\iunit\theta},q^{\frac{1}{2}}\expe^{-\iunit\theta}}{q,-q,-q}{q}{q} }

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


P n ⁡ ( x | q 2 ) = P n ⁡ ( x ; q ) continuous-q-Legendre-polynomial-P 𝑛 𝑥 superscript 𝑞 2 continuous-q-Legendre-Rahman-polynomial-P 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x|q^{2}\right)=P_{n}\!% \left(x;q\right)}}} {\displaystyle \ctsqLegendre{n}@{x}{q^2}=\ctsqLegendreRahman{n}@{x}{q} }
P n ⁡ ( x | q - 1 ) = P n ⁡ ( x | q ) continuous-q-Legendre-polynomial-P 𝑛 𝑥 superscript 𝑞 1 continuous-q-Legendre-polynomial-P 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x|q^{-1}\right)=P_{n}% \!\left(x|q\right)}}} {\displaystyle \ctsqLegendre{n}@{x}{q^{-1}}=\ctsqLegendre{n}@{x}{q} }
P n ⁡ ( x | q ) = q 1 4 ⁢ n ⁢ C n ⁡ ( x ; q 1 2 | q ) continuous-q-Legendre-polynomial-P 𝑛 𝑥 𝑞 superscript 𝑞 1 4 𝑛 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x|q\right)=q^{\frac{1}% {4}n}C_{n}\!\left(x;q^{\frac{1}{2}}\,|\,q\right)}}} {\displaystyle \ctsqLegendre{n}@{x}{q}=q^{\frac{1}{4}n}\ctsqUltra{n}@{x}{q^{\frac{1}{2}}}{q} }

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