Askey-Wilson

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Askey-Wilson

Basic hypergeometric representation

a n ⁢ p n ⁡ ( x ; a , b , c , d | q ) ( a ⁢ b , a ⁢ c , a ⁢ d ; q ) n = \qHyperrphis ⁢ 43 ⁢ @ ⁢ @ ⁢ q - n , a ⁢ b ⁢ c ⁢ d ⁢ q n - 1 , a ⁢ e i ⁢ θ , a ⁢ e - i ⁢ θ ⁢ a ⁢ b , a ⁢ c , a ⁢ d ⁢ q ⁢ q superscript 𝑎 𝑛 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 q-Pochhammer-symbol 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 𝑞 𝑛 \qHyperrphis 43 @ @ superscript 𝑞 𝑛 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 𝑎 imaginary-unit 𝜃 𝑎 imaginary-unit 𝜃 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{a^{n}p_{n}\!\left(x;a,b,c,d\,|% \,q\right)}{\left(ab,ac,ad;q\right)_{n}}{}=\qHyperrphis{4}{3}@@{q^{-n},abcdq^{% n-1},a{\mathrm{e}^{\mathrm{i}\theta}},a{\mathrm{e}^{-\mathrm{i}\theta}}}{ab,ac% ,ad}{q}{q}}}} {\displaystyle \frac{a^n\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ab,ac,ad}{q}{n}} {}=\qHyperrphis{4}{3}@@{q^{-n},abcdq^{n-1},a\expe^{\iunit\theta},a\expe^{-\iunit\theta}}{ab,ac,ad}{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 - x 2 ⁢ p m ⁡ ( x ; a , b , c , d | q ) ⁢ p n ⁡ ( x ; a , b , c , d | q ) ⁢ 𝑑 x = h n ⁢ δ m , n 1 2 superscript subscript 1 1 𝑤 𝑥 1 superscript 𝑥 2 Askey-Wilson-polynomial-p 𝑚 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 differential-d 𝑥 subscript ℎ 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x% )}{\sqrt{1-x^{2}}}p_{m}\!\left(x;a,b,c,d\,|\,q\right)p_{n}\!\left(x;a,b,c,d\,|% \,q\right)\,dx=h_{n}\,\delta_{m,n}}}} {\displaystyle \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\AskeyWilson{m}@{x}{a}{b}{c}{d}{q}\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}\,dx=h_n\,\Kronecker{m}{n} }

Substitution(s): h n = ( a ⁢ b ⁢ c ⁢ d ⁢ q n - 1 ; q ) n ⁢ ( a ⁢ b ⁢ c ⁢ d ⁢ q 2 ⁢ n ; q ) ∞ ( q n + 1 , a ⁢ b ⁢ q n , a ⁢ c ⁢ q n , a ⁢ d ⁢ q n , b ⁢ c ⁢ q n , b ⁢ d ⁢ q n , c ⁢ d ⁢ q n ; q ) ∞ subscript ℎ 𝑛 q-Pochhammer-symbol 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 𝑞 q-Pochhammer-symbol superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 𝑎 𝑐 superscript 𝑞 𝑛 𝑎 𝑑 superscript 𝑞 𝑛 𝑏 𝑐 superscript 𝑞 𝑛 𝑏 𝑑 superscript 𝑞 𝑛 𝑐 𝑑 superscript 𝑞 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle h_{n}=\frac{\left(abcdq^{n-1};q% \right)_{n}\left(abcdq^{2n};q\right)_{\infty}}{\left(q^{n+1},abq^{n},acq^{n},% adq^{n},bcq^{n},bdq^{n},cdq^{n};q\right)_{\infty}}}}} &

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


1 2 ⁢ π ⁢ ∫ - 1 1 w ⁢ ( x ) 1 - x 2 ⁢ p m ⁡ ( x ; a , b , c , d | q ) ⁢ p n ⁡ ( x ; a , b , c , d | q ) ⁢ 𝑑 x + ∑ k 1 < a ⁢ q k ≤ a w k ⁢ p m ⁡ ( x k ; a , b , c , d | q ) ⁢ p n ⁡ ( x k ; a , b , c , d | q ) = h n ⁢ δ m , n 1 2 superscript subscript 1 1 𝑤 𝑥 1 superscript 𝑥 2 Askey-Wilson-polynomial-p 𝑚 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 differential-d 𝑥 subscript 𝑘 1 𝑎 superscript 𝑞 𝑘 𝑎 subscript 𝑤 𝑘 Askey-Wilson-polynomial-p 𝑚 subscript 𝑥 𝑘 𝑎 𝑏 𝑐 𝑑 𝑞 Askey-Wilson-polynomial-p 𝑛 subscript 𝑥 𝑘 𝑎 𝑏 𝑐 𝑑 𝑞 subscript ℎ 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x% )}{\sqrt{1-x^{2}}}p_{m}\!\left(x;a,b,c,d\,|\,q\right)p_{n}\!\left(x;a,b,c,d\,|% \,q\right)\,dx{}+\sum_{\begin{array}[]{c}\scriptstyle k\\ \scriptstyle 1<aq^{k}\leq a\end{array}}w_{k}p_{m}\!\left(x_{k};a,b,c,d\,|\,q% \right)p_{n}\!\left(x_{k};a,b,c,d\,|\,q\right)=h_{n}\,\delta_{m,n}}}} {\displaystyle \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\AskeyWilson{m}@{x}{a}{b}{c}{d}{q}\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}\,dx {}+\sum_{\begin{array}{c}\scriptstyle k\ \scriptstyle 1

Substitution(s): w k = ( a - 2 ; q ) ∞ ( q , a ⁢ b , a ⁢ c , a ⁢ d , a - 1 ⁢ b , a - 1 ⁢ c , a - 1 ⁢ d ; q ) ∞ ⁢ ( 1 - a 2 ⁢ q 2 ⁢ k ) ⁢ ( a 2 , a ⁢ b , a ⁢ c , a ⁢ d ; q ) k ( 1 - a 2 ) ⁢ ( q , a ⁢ b - 1 ⁢ q , a ⁢ c - 1 ⁢ q , a ⁢ d - 1 ⁢ q ; q ) k ⁢ ( q a ⁢ b ⁢ c ⁢ d ) k subscript 𝑤 𝑘 q-Pochhammer-symbol superscript 𝑎 2 𝑞 q-Pochhammer-symbol 𝑞 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 superscript 𝑎 1 𝑏 superscript 𝑎 1 𝑐 superscript 𝑎 1 𝑑 𝑞 1 superscript 𝑎 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol superscript 𝑎 2 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 𝑞 𝑘 1 superscript 𝑎 2 q-Pochhammer-symbol 𝑞 𝑎 superscript 𝑏 1 𝑞 𝑎 superscript 𝑐 1 𝑞 𝑎 superscript 𝑑 1 𝑞 𝑞 𝑘 superscript 𝑞 𝑎 𝑏 𝑐 𝑑 𝑘 {\displaystyle{\displaystyle{\displaystyle w_{k}=\frac{\left(a^{-2};q\right)_{% \infty}}{\left(q,ab,ac,ad,a^{-1}b,a^{-1}c,a^{-1}d;q\right)_{\infty}}{}\frac{(1% -a^{2}q^{2k})\left(a^{2},ab,ac,ad;q\right)_{k}}{(1-a^{2})\left(q,ab^{-1}q,ac^{% -1}q,ad^{-1}q;q\right)_{k}}\left(\frac{q}{abcd}\right)^{k}}}} &

x k = a ⁢ q k + ( a ⁢ q k ) - 1 2 subscript 𝑥 𝑘 𝑎 superscript 𝑞 𝑘 superscript 𝑎 superscript 𝑞 𝑘 1 2 {\displaystyle{\displaystyle{\displaystyle x_{k}=\frac{aq^{k}+\left(aq^{k}% \right)^{-1}}{2}}}} &
h n = ( a ⁢ b ⁢ c ⁢ d ⁢ q n - 1 ; q ) n ⁢ ( a ⁢ b ⁢ c ⁢ d ⁢ q 2 ⁢ n ; q ) ∞ ( q n + 1 , a ⁢ b ⁢ q n , a ⁢ c ⁢ q n , a ⁢ d ⁢ q n , b ⁢ c ⁢ q n , b ⁢ d ⁢ q n , c ⁢ d ⁢ q n ; q ) ∞ subscript ℎ 𝑛 q-Pochhammer-symbol 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 𝑞 q-Pochhammer-symbol superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 𝑎 𝑐 superscript 𝑞 𝑛 𝑎 𝑑 superscript 𝑞 𝑛 𝑏 𝑐 superscript 𝑞 𝑛 𝑏 𝑑 superscript 𝑞 𝑛 𝑐 𝑑 superscript 𝑞 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle h_{n}=\frac{\left(abcdq^{n-1};q% \right)_{n}\left(abcdq^{2n};q\right)_{\infty}}{\left(q^{n+1},abq^{n},acq^{n},% adq^{n},bcq^{n},bdq^{n},cdq^{n};q\right)_{\infty}}}}} &
w ⁢ ( x ) := w ⁢ ( x ; a , b , c , d | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ ⁢ c ⁢ e i ⁢ θ , d ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) ⁢ h ⁢ ( x , c ) ⁢ h ⁢ ( x , d ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 𝑏 𝑐 conditional 𝑑 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑐 imaginary-unit 𝜃 𝑑 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 ℎ 𝑥 𝑏 ℎ 𝑥 𝑐 ℎ 𝑥 𝑑 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c,d|q)=\left|\frac{% \left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}}c{\mathrm{e}^{\mathrm{i}% \theta}},d{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{% h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)h(x,c)h(x,% d)}}}} &
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 ⁡ ( x ) = A n ⁢ p ~ n + 1 ⁡ ( x ) + [ a + a - 1 - ( A n + C n ) ] ⁢ p ~ n ⁡ ( x ) + C n ⁢ p ~ n - 1 ⁡ ( x ) 2 𝑥 Askey-Wilson-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 subscript 𝐴 𝑛 Askey-Wilson-polynomial-normalized-p-tilde 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 delimited-[] 𝑎 superscript 𝑎 1 subscript 𝐴 𝑛 subscript 𝐶 𝑛 Askey-Wilson-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 subscript 𝐶 𝑛 Askey-Wilson-polynomial-normalized-p-tilde 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 {\displaystyle{\displaystyle{\displaystyle 2x{\tilde{p}}_{n}\!\left(x\right)=A% _{n}{\tilde{p}}_{n+1}\!\left(x\right)+\left[a+a^{-1}-\left(A_{n}+C_{n}\right)% \right]{\tilde{p}}_{n}\!\left(x\right)+C_{n}{\tilde{p}}_{n-1}\!\left(x\right)}}} {\displaystyle 2x\normAskeyWilsonptilde{n}@@{x}{a}{b}{c}{d}{q}=A_n\normAskeyWilsonptilde{n+1}@@{x}{a}{b}{c}{d}{q}+\left[a+a^{-1}-\left(A_n+C_n\right)\right]\normAskeyWilsonptilde{n}@@{x}{a}{b}{c}{d}{q}+C_n\normAskeyWilsonptilde{n-1}@@{x}{a}{b}{c}{d}{q} }

Substitution(s): C n = a ⁢ ( 1 - q n ) ⁢ ( 1 - b ⁢ c ⁢ q n - 1 ) ⁢ ( 1 - b ⁢ d ⁢ q n - 1 ) ⁢ ( 1 - c ⁢ d ⁢ q n - 1 ) ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q 2 ⁢ n - 2 ) ⁢ ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q 2 ⁢ n - 1 ) subscript 𝐶 𝑛 𝑎 1 superscript 𝑞 𝑛 1 𝑏 𝑐 superscript 𝑞 𝑛 1 1 𝑏 𝑑 superscript 𝑞 𝑛 1 1 𝑐 𝑑 superscript 𝑞 𝑛 1 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 2 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{a(1-q^{n})(1-bcq^{n-1})% (1-bdq^{n-1})(1-cdq^{n-1})}{(1-abcdq^{2n-2})(1-abcdq^{2n-1})}}}} &
A n = ( 1 - a ⁢ b ⁢ q n ) ⁢ ( 1 - a ⁢ c ⁢ q n ) ⁢ ( 1 - a ⁢ d ⁢ q n ) ⁢ ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q n - 1 ) a ⁢ ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q 2 ⁢ n - 1 ) ⁢ ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q 2 ⁢ n ) subscript 𝐴 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 1 𝑎 𝑐 superscript 𝑞 𝑛 1 𝑎 𝑑 superscript 𝑞 𝑛 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 𝑎 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 1 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-abq^{n})(1-acq^{n})(% 1-adq^{n})(1-abcdq^{n-1})}{a(1-abcdq^{2n-1})(1-abcdq^{2n})}}}}


p ~ n ⁡ ( x ) := p ~ n ⁡ ( x ; a , b , c , d | q ) = a n ⁢ p n ⁡ ( x ; a , b , c , d | q ) ( a ⁢ b , a ⁢ c , a ⁢ d ; q ) n assign Askey-Wilson-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 Askey-Wilson-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 superscript 𝑎 𝑛 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 q-Pochhammer-symbol 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle{\tilde{p}}_{n}\!\left(x\right):={% \tilde{p}}_{n}\!\left(x;a,b,c,d\,|\,q\right)=\frac{a^{n}p_{n}\!\left(x;a,b,c,d% \,|\,q\right)}{\left(ab,ac,ad;q\right)_{n}}}}} {\displaystyle \normAskeyWilsonptilde{n}@@{x}{a}{b}{c}{d}{q}:=\normAskeyWilsonptilde{n}@{x}{a}{b}{c}{d}{q}=\frac{a^n\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ab,ac,ad}{q}{n}} }

Monic recurrence relation

x ⁢ p ^ n ⁡ ( x ) = p ^ n + 1 ⁡ ( x ) + 1 2 ⁢ [ a + a - 1 - ( A n + C n ) ] ⁢ p ^ n ⁡ ( x ) + 1 4 ⁢ A n - 1 ⁢ C n ⁢ p ^ n - 1 ⁡ ( x ) 𝑥 Askey-Wilson-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 Askey-Wilson-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 1 2 delimited-[] 𝑎 superscript 𝑎 1 subscript 𝐴 𝑛 subscript 𝐶 𝑛 Askey-Wilson-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 1 4 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 Askey-Wilson-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{p}}_{n}\!\left(x\right)=% {\widehat{p}}_{n+1}\!\left(x\right)+\frac{1}{2}\left[a+a^{-1}-(A_{n}+C_{n})% \right]{\widehat{p}}_{n}\!\left(x\right)+\frac{1}{4}A_{n-1}C_{n}{\widehat{p}}_% {n-1}\!\left(x\right)}}} {\displaystyle x\monicAskeyWilson{n}@@{x}{a}{b}{c}{d}{q}=\monicAskeyWilson{n+1}@@{x}{a}{b}{c}{d}{q}+\frac{1}{2}\left[a+a^{-1}-(A_n+C_n)\right]\monicAskeyWilson{n}@@{x}{a}{b}{c}{d}{q}+ \frac{1}{4}A_{n-1}C_n\monicAskeyWilson{n-1}@@{x}{a}{b}{c}{d}{q} }

Substitution(s): C n = a ⁢ ( 1 - q n ) ⁢ ( 1 - b ⁢ c ⁢ q n - 1 ) ⁢ ( 1 - b ⁢ d ⁢ q n - 1 ) ⁢ ( 1 - c ⁢ d ⁢ q n - 1 ) ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q 2 ⁢ n - 2 ) ⁢ ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q 2 ⁢ n - 1 ) subscript 𝐶 𝑛 𝑎 1 superscript 𝑞 𝑛 1 𝑏 𝑐 superscript 𝑞 𝑛 1 1 𝑏 𝑑 superscript 𝑞 𝑛 1 1 𝑐 𝑑 superscript 𝑞 𝑛 1 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 2 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{a(1-q^{n})(1-bcq^{n-1})% (1-bdq^{n-1})(1-cdq^{n-1})}{(1-abcdq^{2n-2})(1-abcdq^{2n-1})}}}} &
A n = ( 1 - a ⁢ b ⁢ q n ) ⁢ ( 1 - a ⁢ c ⁢ q n ) ⁢ ( 1 - a ⁢ d ⁢ q n ) ⁢ ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q n - 1 ) a ⁢ ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q 2 ⁢ n - 1 ) ⁢ ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q 2 ⁢ n ) subscript 𝐴 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 1 𝑎 𝑐 superscript 𝑞 𝑛 1 𝑎 𝑑 superscript 𝑞 𝑛 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 𝑎 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 1 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-abq^{n})(1-acq^{n})(% 1-adq^{n})(1-abcdq^{n-1})}{a(1-abcdq^{2n-1})(1-abcdq^{2n})}}}}


p n ⁡ ( x ; a , b , c , d | q ) = 2 n ⁢ ( a ⁢ b ⁢ c ⁢ d ⁢ q n - 1 ; q ) n ⁢ p ^ n ⁡ ( x ) Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 superscript 2 𝑛 q-Pochhammer-symbol 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 𝑞 𝑛 Askey-Wilson-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b,c,d\,|\,q\right)% =2^{n}\left(abcdq^{n-1};q\right)_{n}{\widehat{p}}_{n}\!\left(x\right)}}} {\displaystyle \AskeyWilson{n}@{x}{a}{b}{c}{d}{q}=2^n\qPochhammer{abcdq^{n-1}}{q}{n}\monicAskeyWilson{n}@@{x}{a}{b}{c}{d}{q} }

q-Difference equation

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

Substitution(s): y ⁢ ( x ) = p n ⁡ ( x ; a , b , c , d | q ) 𝑦 𝑥 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=p_{n}\!\left(x;a,b,c,d\,|\,q% \right)}}} &

λ n = 4 ⁢ q - n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q n - 1 ) subscript 𝜆 𝑛 4 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\lambda_{n}=4q^{-n+1}(1-q^{n})(1-% abcdq^{n-1})}}} &
w ~ ⁢ ( x ; a , b , c , d | q ) := w ⁢ ( x ; a , b , c , d | q ) 1 - x 2 assign ~ 𝑤 𝑥 𝑎 𝑏 𝑐 conditional 𝑑 𝑞 𝑤 𝑥 𝑎 𝑏 𝑐 conditional 𝑑 𝑞 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b,c,d|q):=\frac{w(x;% a,b,c,d|q)}{\sqrt{1-x^{2}}}}}} &
w ⁢ ( x ) := w ⁢ ( x ; a , b , c , d | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ ⁢ c ⁢ e i ⁢ θ , d ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) ⁢ h ⁢ ( x , c ) ⁢ h ⁢ ( x , d ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 𝑏 𝑐 conditional 𝑑 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑐 imaginary-unit 𝜃 𝑑 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 ℎ 𝑥 𝑏 ℎ 𝑥 𝑐 ℎ 𝑥 𝑑 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c,d|q)=\left|\frac{% \left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}}c{\mathrm{e}^{\mathrm{i}% \theta}},d{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{% h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)h(x,c)h(x,% d)}}}} &
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}}}


q - n ⁢ ( 1 - q n ) ⁢ ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q n - 1 ) ⁢ 𝒫 n ⁢ ( z ) = A ⁢ ( z ) ⁢ 𝒫 n ⁢ ( q ⁢ z ) - [ A ⁢ ( z ) + A ⁢ ( z - 1 ) ] ⁢ 𝒫 n ⁢ ( z ) + A ⁢ ( z - 1 ) ⁢ 𝒫 n ⁢ ( q - 1 ⁢ z ) superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 subscript 𝒫 𝑛 𝑧 𝐴 𝑧 subscript 𝒫 𝑛 𝑞 𝑧 delimited-[] 𝐴 𝑧 𝐴 superscript 𝑧 1 subscript 𝒫 𝑛 𝑧 𝐴 superscript 𝑧 1 subscript 𝒫 𝑛 superscript 𝑞 1 𝑧 {\displaystyle{\displaystyle{\displaystyle q^{-n}(1-q^{n})(1-abcdq^{n-1}){% \mathcal{P}}_{n}(z){}=A(z){\mathcal{P}}_{n}(qz)-\left[A(z)+A(z^{-1})\right]{% \mathcal{P}}_{n}(z)+A(z^{-1}){\mathcal{P}}_{n}(q^{-1}z)}}} {\displaystyle q^{-n}(1-q^n)(1-abcdq^{n-1}){\mathcal P}_n(z) {}=A(z){\mathcal P}_n(qz)-\left[A(z)+A(z^{-1})\right]{\mathcal P}_n(z)+A(z^{-1}){\mathcal P}_n(q^{-1}z) }

Substitution(s): A ⁢ ( z ) = ( 1 - a ⁢ z ) ⁢ ( 1 - b ⁢ z ) ⁢ ( 1 - c ⁢ z ) ⁢ ( 1 - d ⁢ z ) ( 1 - z 2 ) ⁢ ( 1 - q ⁢ z 2 ) 𝐴 𝑧 1 𝑎 𝑧 1 𝑏 𝑧 1 𝑐 𝑧 1 𝑑 𝑧 1 superscript 𝑧 2 1 𝑞 superscript 𝑧 2 {\displaystyle{\displaystyle{\displaystyle A(z)=\frac{(1-az)(1-bz)(1-cz)(1-dz)% }{(1-z^{2})(1-qz^{2})}}}} &
𝒫 n ⁢ ( z ) := ( a ⁢ b , a ⁢ c , a ⁢ d ; q ) n a n ⁢ \qHyperrphis ⁢ 43 ⁢ @ ⁢ @ ⁢ q - n , a ⁢ b ⁢ c ⁢ d ⁢ q n - 1 , a ⁢ z , a ⁢ z - 1 ⁢ a ⁢ b , a ⁢ c , a ⁢ d ⁢ q ⁢ q assign subscript 𝒫 𝑛 𝑧 q-Pochhammer-symbol 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 𝑞 𝑛 superscript 𝑎 𝑛 \qHyperrphis 43 @ @ superscript 𝑞 𝑛 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 𝑎 𝑧 𝑎 superscript 𝑧 1 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle{\mathcal{P}}_{n}(z):=\frac{\left(ab% ,ac,ad;q\right)_{n}}{a^{n}}\,\qHyperrphis{4}{3}@@{q^{-n},abcdq^{n-1},az,az^{-1% }}{ab,ac,ad}{q}{q}}}}


Forward shift operator

δ q ⁢ p n ⁡ ( x ; a , b , c , d | q ) = - q - 1 2 ⁢ n ⁢ ( 1 - q n ) ⁢ ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q n - 1 ) ⁢ ( e i ⁢ θ - e - i ⁢ θ ) ⁢ p n - 1 ⁡ ( x ; a ⁢ q 1 2 , b ⁢ q 1 2 , c ⁢ q 1 2 , d ⁢ q 1 2 | q ) subscript 𝛿 𝑞 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 superscript 𝑞 1 2 𝑛 1 superscript 𝑞 𝑛 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 imaginary-unit 𝜃 imaginary-unit 𝜃 Askey-Wilson-polynomial-p 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑏 superscript 𝑞 1 2 𝑐 superscript 𝑞 1 2 𝑑 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}p_{n}\!\left(x;a,b,c,d\,|% \,q\right)=-q^{-\frac{1}{2}n}(1-q^{n})(1-abcdq^{n-1})({\mathrm{e}^{\mathrm{i}% \theta}}-{\mathrm{e}^{-\mathrm{i}\theta}}){}p_{n-1}\!\left(x;aq^{\frac{1}{2}},% bq^{\frac{1}{2}},cq^{\frac{1}{2}},dq^{\frac{1}{2}}\,|\,q\right)}}} {\displaystyle \delta_q \AskeyWilson{n}@{x}{a}{b}{c}{d}{q}=-q^{-\frac{1}{2}n}(1-q^n)(1-abcdq^{n-1})(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) {} \AskeyWilson{n-1}@{x}{aq^{\frac{1}{2}}}{bq^{\frac{1}{2}}}{cq^{\frac{1}{2}}}{dq^{\frac{1}{2}}}{q} }
D q ⁢ p n ⁡ ( x ; a , b , c , d | q ) = 2 ⁢ q - 1 2 ⁢ ( n - 1 ) ⁢ ( 1 - q n ) ⁢ ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q n - 1 ) 1 - q ⁢ p n - 1 ⁡ ( x ; a ⁢ q 1 2 , b ⁢ q 1 2 , c ⁢ q 1 2 , d ⁢ q 1 2 | q ) subscript 𝐷 𝑞 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 2 superscript 𝑞 1 2 𝑛 1 1 superscript 𝑞 𝑛 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 1 𝑞 Askey-Wilson-polynomial-p 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑏 superscript 𝑞 1 2 𝑐 superscript 𝑞 1 2 𝑑 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle D_{q}p_{n}\!\left(x;a,b,c,d\,|\,q% \right)=2q^{-\frac{1}{2}(n-1)}\frac{(1-q^{n})(1-abcdq^{n-1})}{1-q}{}p_{n-1}\!% \left(x;aq^{\frac{1}{2}},bq^{\frac{1}{2}},cq^{\frac{1}{2}},dq^{\frac{1}{2}}\,|% \,q\right)}}} {\displaystyle D_q \AskeyWilson{n}@{x}{a}{b}{c}{d}{q}=2q^{-\frac{1}{2}(n-1)}\frac{(1-q^n)(1-abcdq^{n-1})}{1-q} {} \AskeyWilson{n-1}@{x}{aq^{\frac{1}{2}}}{bq^{\frac{1}{2}}}{cq^{\frac{1}{2}}}{dq^{\frac{1}{2}}}{q} }

Backward shift operator

δ q ⁢ [ w ~ ⁢ ( x ; a , b , c , d | q ) ⁢ p n ⁡ ( x ; a , b , c , d | q ) ] = q - 1 2 ⁢ ( n + 1 ) ⁢ ( e i ⁢ θ - e - i ⁢ θ ) ⁢ w ~ ⁢ ( x ; a ⁢ q - 1 2 , b ⁢ q - 1 2 , c ⁢ q - 1 2 , d ⁢ q - 1 2 | q ) ⁢ p n + 1 ⁡ ( x ; a ⁢ q - 1 2 , b ⁢ q - 1 2 , c ⁢ q - 1 2 , d ⁢ q - 1 2 | q ) subscript 𝛿 𝑞 delimited-[] ~ 𝑤 𝑥 𝑎 𝑏 𝑐 conditional 𝑑 𝑞 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 superscript 𝑞 1 2 𝑛 1 imaginary-unit 𝜃 imaginary-unit 𝜃 ~ 𝑤 𝑥 𝑎 superscript 𝑞 1 2 𝑏 superscript 𝑞 1 2 𝑐 superscript 𝑞 1 2 conditional 𝑑 superscript 𝑞 1 2 𝑞 Askey-Wilson-polynomial-p 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑏 superscript 𝑞 1 2 𝑐 superscript 𝑞 1 2 𝑑 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}\left[{\tilde{w}}(x;a,b,c,% d|q)p_{n}\!\left(x;a,b,c,d\,|\,q\right)\right]{}=q^{-\frac{1}{2}(n+1)}({% \mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e}^{-\mathrm{i}\theta}}){\tilde{w}}(x;% aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}},dq^{-\frac{1}{2}}|q){}p_% {n+1}\!\left(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}},dq^{-% \frac{1}{2}}\,|\,q\right)}}} {\displaystyle \delta_q\left[{\tilde w}(x;a,b,c,d|q)\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}\right] {}=q^{-\frac{1}{2}(n+1)}(\expe^{\iunit\theta}-\expe^{-\iunit\theta}){\tilde w}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}},dq^{-\frac{1}{2}}|q) {} \AskeyWilson{n+1}@{x}{aq^{-\frac{1}{2}}}{bq^{-\frac{1}{2}}}{cq^{-\frac{1}{2}}}{dq^{-\frac{1}{2}}}{q} }

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

w ⁢ ( x ) := w ⁢ ( x ; a , b , c , d | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ ⁢ c ⁢ e i ⁢ θ , d ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) ⁢ h ⁢ ( x , c ) ⁢ h ⁢ ( x , d ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 𝑏 𝑐 conditional 𝑑 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑐 imaginary-unit 𝜃 𝑑 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 ℎ 𝑥 𝑏 ℎ 𝑥 𝑐 ℎ 𝑥 𝑑 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c,d|q)=\left|\frac{% \left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}}c{\mathrm{e}^{\mathrm{i}% \theta}},d{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{% h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)h(x,c)h(x,% d)}}}} &
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 ; a , b , c , d | q ) ⁢ p n ⁡ ( x ; a , b , c , d | q ) ] = - 2 ⁢ q - 1 2 ⁢ n 1 - q ⁢ w ~ ⁢ ( x ; a ⁢ q - 1 2 , b ⁢ q - 1 2 , c ⁢ q - 1 2 , d ⁢ q - 1 2 | q ) ⁢ p n + 1 ⁡ ( x ; a ⁢ q - 1 2 , b ⁢ q - 1 2 , c ⁢ q - 1 2 , d ⁢ q - 1 2 | q ) subscript 𝐷 𝑞 delimited-[] ~ 𝑤 𝑥 𝑎 𝑏 𝑐 conditional 𝑑 𝑞 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 2 superscript 𝑞 1 2 𝑛 1 𝑞 ~ 𝑤 𝑥 𝑎 superscript 𝑞 1 2 𝑏 superscript 𝑞 1 2 𝑐 superscript 𝑞 1 2 conditional 𝑑 superscript 𝑞 1 2 𝑞 Askey-Wilson-polynomial-p 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑏 superscript 𝑞 1 2 𝑐 superscript 𝑞 1 2 𝑑 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle D_{q}\left[{\tilde{w}}(x;a,b,c,d|q)% p_{n}\!\left(x;a,b,c,d\,|\,q\right)\right]{}=-\frac{2q^{-\frac{1}{2}n}}{1-q}{% \tilde{w}}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}},dq^{-\frac{% 1}{2}}|q){}p_{n+1}\!\left(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{% 2}},dq^{-\frac{1}{2}}\,|\,q\right)}}} {\displaystyle D_q\left[{\tilde w}(x;a,b,c,d|q)\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}\right] {}=-\frac{2q^{-\frac{1}{2}n}}{1-q}{\tilde w}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}},dq^{-\frac{1}{2}}|q) {} \AskeyWilson{n+1}@{x}{aq^{-\frac{1}{2}}}{bq^{-\frac{1}{2}}}{cq^{-\frac{1}{2}}}{dq^{-\frac{1}{2}}}{q} }

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

w ⁢ ( x ) := w ⁢ ( x ; a , b , c , d | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ ⁢ c ⁢ e i ⁢ θ , d ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) ⁢ h ⁢ ( x , c ) ⁢ h ⁢ ( x , d ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 𝑏 𝑐 conditional 𝑑 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑐 imaginary-unit 𝜃 𝑑 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 ℎ 𝑥 𝑏 ℎ 𝑥 𝑐 ℎ 𝑥 𝑑 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c,d|q)=\left|\frac{% \left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}}c{\mathrm{e}^{\mathrm{i}% \theta}},d{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{% h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)h(x,c)h(x,% d)}}}} &
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 ; a , b , c , d | q ) ⁢ p n ⁡ ( x ; a , b , c , d | q ) = ( q - 1 2 ) n ⁢ q 1 4 ⁢ n ⁢ ( n - 1 ) ⁢ ( D q ) n ⁢ [ w ~ ⁢ ( x ; a ⁢ q 1 2 ⁢ n , b ⁢ q 1 2 ⁢ n , c ⁢ q 1 2 ⁢ n , d ⁢ q 1 2 ⁢ n | q ) ] ~ 𝑤 𝑥 𝑎 𝑏 𝑐 conditional 𝑑 𝑞 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 superscript 𝑞 1 2 𝑛 superscript 𝑞 1 4 𝑛 𝑛 1 superscript subscript 𝐷 𝑞 𝑛 delimited-[] ~ 𝑤 𝑥 𝑎 superscript 𝑞 1 2 𝑛 𝑏 superscript 𝑞 1 2 𝑛 𝑐 superscript 𝑞 1 2 𝑛 conditional 𝑑 superscript 𝑞 1 2 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b,c,d|q)p_{n}\!\left% (x;a,b,c,d\,|\,q\right){}=\left(\frac{q-1}{2}\right)^{n}q^{\frac{1}{4}n(n-1)}% \left(D_{q}\right)^{n}\left[{\tilde{w}}(x;aq^{\frac{1}{2}n},bq^{\frac{1}{2}n},% cq^{\frac{1}{2}n},dq^{\frac{1}{2}n}|q)\right]}}} {\displaystyle {\tilde w}(x;a,b,c,d|q)\AskeyWilson{n}@{x}{a}{b}{c}{d}{q} {}=\left(\frac{q-1}{2}\right)^nq^{\frac{1}{4}n(n-1)}\left(D_q\right)^n \left[{\tilde w}(x;aq^{\frac{1}{2}n},bq^{\frac{1}{2}n},cq^{\frac{1}{2}n},dq^{\frac{1}{2}n}|q)\right] }

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

w ⁢ ( x ) := w ⁢ ( x ; a , b , c , d | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ ⁢ c ⁢ e i ⁢ θ , d ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) ⁢ h ⁢ ( x , c ) ⁢ h ⁢ ( x , d ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 𝑏 𝑐 conditional 𝑑 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑐 imaginary-unit 𝜃 𝑑 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 ℎ 𝑥 𝑏 ℎ 𝑥 𝑐 ℎ 𝑥 𝑑 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c,d|q)=\left|\frac{% \left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}}c{\mathrm{e}^{\mathrm{i}% \theta}},d{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{% h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)h(x,c)h(x,% d)}}}} &
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 ⁢ @ ⁢ @ ⁢ a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ ⁢ a ⁢ b ⁢ q ⁢ e - i ⁢ θ ⁢ t ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ c ⁢ e - i ⁢ θ , d ⁢ e - i ⁢ θ ⁢ c ⁢ d ⁢ q ⁢ e i ⁢ θ ⁢ t = ∑ n = 0 ∞ p n ⁡ ( x ; a , b , c , d | q ) ( a ⁢ b , c ⁢ d , q ; q ) n ⁢ t n \qHyperrphis 21 @ @ 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑎 𝑏 𝑞 imaginary-unit 𝜃 𝑡 \qHyperrphis 21 @ @ 𝑐 imaginary-unit 𝜃 𝑑 imaginary-unit 𝜃 𝑐 𝑑 𝑞 imaginary-unit 𝜃 𝑡 superscript subscript 𝑛 0 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 q-Pochhammer-symbol 𝑎 𝑏 𝑐 𝑑 𝑞 𝑞 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{a{\mathrm{e}^{% \mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}}}{ab}{q}{{\mathrm{e}^{-% \mathrm{i}\theta}}t}\ \qHyperrphis{2}{1}@@{c{\mathrm{e}^{-\mathrm{i}\theta}},d% {\mathrm{e}^{-\mathrm{i}\theta}}}{cd}{q}{{\mathrm{e}^{\mathrm{i}\theta}}t}{}=% \sum_{n=0}^{\infty}\frac{p_{n}\!\left(x;a,b,c,d\,|\,q\right)}{\left(ab,cd,q;q% \right)_{n}}t^{n}}}} {\displaystyle \qHyperrphis{2}{1}@@{a\expe^{\iunit\theta},b\expe^{\iunit\theta}}{ab}{q}{\expe^{-\iunit\theta}t}\ \qHyperrphis{2}{1}@@{c\expe^{-\iunit\theta},d\expe^{-\iunit\theta}}{cd}{q}{\expe^{\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ab,cd,q}{q}{n}}t^n }

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


\qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ a ⁢ e i ⁢ θ , c ⁢ e i ⁢ θ ⁢ a ⁢ c ⁢ q ⁢ e - i ⁢ θ ⁢ t ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ b ⁢ e - i ⁢ θ , d ⁢ e - i ⁢ θ ⁢ b ⁢ d ⁢ q ⁢ e i ⁢ θ ⁢ t = ∑ n = 0 ∞ p n ⁡ ( x ; a , b , c , d | q ) ( a ⁢ c , b ⁢ d , q ; q ) n ⁢ t n \qHyperrphis 21 @ @ 𝑎 imaginary-unit 𝜃 𝑐 imaginary-unit 𝜃 𝑎 𝑐 𝑞 imaginary-unit 𝜃 𝑡 \qHyperrphis 21 @ @ 𝑏 imaginary-unit 𝜃 𝑑 imaginary-unit 𝜃 𝑏 𝑑 𝑞 imaginary-unit 𝜃 𝑡 superscript subscript 𝑛 0 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 q-Pochhammer-symbol 𝑎 𝑐 𝑏 𝑑 𝑞 𝑞 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{a{\mathrm{e}^{% \mathrm{i}\theta}},c{\mathrm{e}^{\mathrm{i}\theta}}}{ac}{q}{{\mathrm{e}^{-% \mathrm{i}\theta}}t}\ \qHyperrphis{2}{1}@@{b{\mathrm{e}^{-\mathrm{i}\theta}},d% {\mathrm{e}^{-\mathrm{i}\theta}}}{bd}{q}{{\mathrm{e}^{\mathrm{i}\theta}}t}{}=% \sum_{n=0}^{\infty}\frac{p_{n}\!\left(x;a,b,c,d\,|\,q\right)}{\left(ac,bd,q;q% \right)_{n}}t^{n}}}} {\displaystyle \qHyperrphis{2}{1}@@{a\expe^{\iunit\theta},c\expe^{\iunit\theta}}{ac}{q}{\expe^{-\iunit\theta}t}\ \qHyperrphis{2}{1}@@{b\expe^{-\iunit\theta},d\expe^{-\iunit\theta}}{bd}{q}{\expe^{\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ac,bd,q}{q}{n}}t^n }

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


\qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ a ⁢ e i ⁢ θ , d ⁢ e i ⁢ θ ⁢ a ⁢ d ⁢ q ⁢ e - i ⁢ θ ⁢ t ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ b ⁢ e - i ⁢ θ , c ⁢ e - i ⁢ θ ⁢ b ⁢ c ⁢ q ⁢ e i ⁢ θ ⁢ t = ∑ n = 0 ∞ p n ⁡ ( x ; a , b , c , d | q ) ( a ⁢ d , b ⁢ c , q ; q ) n ⁢ t n \qHyperrphis 21 @ @ 𝑎 imaginary-unit 𝜃 𝑑 imaginary-unit 𝜃 𝑎 𝑑 𝑞 imaginary-unit 𝜃 𝑡 \qHyperrphis 21 @ @ 𝑏 imaginary-unit 𝜃 𝑐 imaginary-unit 𝜃 𝑏 𝑐 𝑞 imaginary-unit 𝜃 𝑡 superscript subscript 𝑛 0 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 q-Pochhammer-symbol 𝑎 𝑑 𝑏 𝑐 𝑞 𝑞 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{a{\mathrm{e}^{% \mathrm{i}\theta}},d{\mathrm{e}^{\mathrm{i}\theta}}}{ad}{q}{{\mathrm{e}^{-% \mathrm{i}\theta}}t}\ \qHyperrphis{2}{1}@@{b{\mathrm{e}^{-\mathrm{i}\theta}},c% {\mathrm{e}^{-\mathrm{i}\theta}}}{bc}{q}{{\mathrm{e}^{\mathrm{i}\theta}}t}{}=% \sum_{n=0}^{\infty}\frac{p_{n}\!\left(x;a,b,c,d\,|\,q\right)}{\left(ad,bc,q;q% \right)_{n}}t^{n}}}} {\displaystyle \qHyperrphis{2}{1}@@{a\expe^{\iunit\theta},d\expe^{\iunit\theta}}{ad}{q}{\expe^{-\iunit\theta}t}\ \qHyperrphis{2}{1}@@{b\expe^{-\iunit\theta},c\expe^{-\iunit\theta}}{bc}{q}{\expe^{\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ad,bc,q}{q}{n}}t^n }

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


Limit relations

Askey-Wilson polynomial to Continuous dual q-Hahn polynomial

p n ⁡ ( x ; a , b , c , 0 | q ) = p n ⁡ ( x ; a , b , c | q ) Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 0 𝑞 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b,c,0\,|\,q\right)% =p_{n}\!\left(x;a,b,c|q\right)}}} {\displaystyle \AskeyWilson{n}@{x}{a}{b}{c}{0}{q}=\ctsdualqHahn{n}@{x}{a}{b}{c}{q} }

Askey-Wilson polynomial to Continuous q-Hahn polynomial

p n ⁡ ( cos ⁡ ( θ + ϕ ) ; a ⁢ e i ⁢ ϕ , b ⁢ e i ⁢ ϕ , c ⁢ e - i ⁢ ϕ , d ⁢ e - i ⁢ ϕ | q ) = p n ⁡ ( cos ⁡ ( θ + ϕ ) ; a , b , c , d ; q ) Askey-Wilson-polynomial-p 𝑛 𝜃 italic-ϕ 𝑎 imaginary-unit italic-ϕ 𝑏 imaginary-unit italic-ϕ 𝑐 imaginary-unit italic-ϕ 𝑑 imaginary-unit italic-ϕ 𝑞 continuous-q-Hahn-polynomial-p 𝑛 𝜃 italic-ϕ 𝑎 𝑏 𝑐 𝑑 𝑞 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(\cos\left(\theta+\phi% \right);a{\mathrm{e}^{\mathrm{i}\phi}},b{\mathrm{e}^{\mathrm{i}\phi}},c{% \mathrm{e}^{-\mathrm{i}\phi}},d{\mathrm{e}^{-\mathrm{i}\phi}}\,|\,q\right)=p_{% n}\!\left(\cos\left(\theta+\phi\right);a,b,c,d;q\right)}}} {\displaystyle \AskeyWilson{n}@{\cos@{\theta+\phi}}{a\expe^{\iunit\phi}}{b\expe^{\iunit\phi}}{c\expe^{-\iunit\phi}}{d\expe^{-\iunit\phi}}{q}=\ctsqHahn{n}@{\cos@{\theta+\phi}}{a}{b}{c}{d}{q} }

Askey-Wilson polynomial to Big q-Jacobi polynomial

p ~ n ⁡ ( x ; a , b , c , d | q ) = a n ⁢ p n ⁡ ( x ; a , b , c , d | q ) ( a ⁢ b , a ⁢ c , a ⁢ d ; q ) n Askey-Wilson-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 superscript 𝑎 𝑛 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 q-Pochhammer-symbol 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle{\tilde{p}}_{n}\!\left(x;a,b,c,d\,|% \,q\right)=\frac{a^{n}p_{n}\!\left(x;a,b,c,d\,|\,q\right)}{\left(ab,ac,ad;q% \right)_{n}}}}} {\displaystyle \normAskeyWilsonptilde{n}@{x}{a}{b}{c}{d}{q}=\frac{a^n\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ab,ac,ad}{q}{n}} }
lim a → 0 ⁡ p ~ n ⁡ ( 1 2 ⁢ a - 1 ⁢ x ; a , a - 1 ⁢ α ⁢ q , a - 1 ⁢ γ ⁢ q , a ⁢ β ⁢ γ - 1 | q ) = P n ⁡ ( x ; α , β , γ ; q ) subscript → 𝑎 0 Askey-Wilson-polynomial-normalized-p-tilde 𝑛 1 2 superscript 𝑎 1 𝑥 𝑎 superscript 𝑎 1 𝛼 𝑞 superscript 𝑎 1 𝛾 𝑞 𝑎 𝛽 superscript 𝛾 1 𝑞 big-q-Jacobi-polynomial-P 𝑛 𝑥 𝛼 𝛽 𝛾 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow 0}{\tilde{p}}_{n}% \!\left(\textstyle\frac{1}{2}a^{-1}x;a,a^{-1}\alpha q,a^{-1}\gamma q,a\beta% \gamma^{-1}\,|\,q\right)=P_{n}\!\left(x;\alpha,\beta,\gamma;q\right)}}} {\displaystyle \lim_{a\rightarrow 0}\normAskeyWilsonptilde{n}@{\textstyle\frac{1}{2}a^{-1}x}{a}{a^{-1}\alpha q}{ a^{-1}\gamma q}{a\beta \gamma^{-1}}{q}=\bigqJacobi{n}@{x}{\alpha}{\beta}{\gamma}{q} }

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} }

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} }

Askey-Wilson polynomial to Wilson polynomial

lim q → 1 ⁡ p n ⁡ ( 1 2 ⁢ ( q i ⁢ x + q - i ⁢ x ) ; q a , q b , q c , q d | q ) ( 1 - q ) 3 ⁢ n = W n ⁡ ( x 2 ; a , b , c , d ) subscript → 𝑞 1 Askey-Wilson-polynomial-p 𝑛 1 2 superscript 𝑞 imaginary-unit 𝑥 superscript 𝑞 imaginary-unit 𝑥 superscript 𝑞 𝑎 superscript 𝑞 𝑏 superscript 𝑞 𝑐 superscript 𝑞 𝑑 𝑞 superscript 1 𝑞 3 𝑛 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{p_{n}\!% \left(\frac{1}{2}\left(q^{\mathrm{i}x}+q^{-\mathrm{i}x}\right);q^{a},q^{b},q^{% c},q^{d}\,|\,q\right)}{(1-q)^{3n}}=W_{n}\!\left(x^{2};a,b,c,d\right)}}} {\displaystyle \lim_{q\rightarrow 1}\frac{\AskeyWilson{n}@{\frac{1}{2}\left(q^{\iunit x}+q^{-\iunit x}\right)}{q^a}{q^b}{q^c}{q^d}{q}}{(1-q)^{3n}}=\Wilson{n}@{x^2}{a}{b}{c}{d} }

Remarks

R n ⁡ ( μ ⁢ ( x ) ; α , β , γ , δ | q ) = ( γ ⁢ δ ⁢ q ) 1 2 ⁢ n ⁢ p n ⁡ ( ν ⁢ ( x ) ; γ 1 2 ⁢ δ 1 2 ⁢ q 1 2 , α ⁢ γ - 1 2 ⁢ δ - 1 2 ⁢ q 1 2 , β ⁢ γ - 1 2 ⁢ δ 1 2 ⁢ q 1 2 , γ 1 2 ⁢ δ - 1 2 ⁢ q 1 2 | q ) ( α ⁢ q , β ⁢ δ ⁢ q , γ ⁢ q ; q ) n q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 superscript 𝛾 𝛿 𝑞 1 2 𝑛 Askey-Wilson-polynomial-p 𝑛 𝜈 𝑥 superscript 𝛾 1 2 superscript 𝛿 1 2 superscript 𝑞 1 2 𝛼 superscript 𝛾 1 2 superscript 𝛿 1 2 superscript 𝑞 1 2 𝛽 superscript 𝛾 1 2 superscript 𝛿 1 2 superscript 𝑞 1 2 superscript 𝛾 1 2 superscript 𝛿 1 2 superscript 𝑞 1 2 𝑞 q-Pochhammer-symbol 𝛼 𝑞 𝛽 𝛿 𝑞 𝛾 𝑞 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\alpha,\beta,% \gamma,\delta\,|\,q\right){}=\frac{(\gamma\delta q)^{\frac{1}{2}n}p_{n}\!\left% (\nu(x);\gamma^{\frac{1}{2}}\delta^{\frac{1}{2}}q^{\frac{1}{2}},\alpha\gamma^{% -\frac{1}{2}}\delta^{-\frac{1}{2}}q^{\frac{1}{2}},\beta\gamma^{-\frac{1}{2}}% \delta^{\frac{1}{2}}q^{\frac{1}{2}},\gamma^{\frac{1}{2}}\delta^{-\frac{1}{2}}q% ^{\frac{1}{2}}\,|\,q\right)}{\left(\alpha q,\beta\delta q,\gamma q;q\right)_{n% }}}}} {\displaystyle \qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q} {}=\frac{(\gamma\delta q)^{\frac{1}{2}n} \AskeyWilson{n}@{\nu(x)}{\gamma^{\frac{1}{2}}\delta^{\frac{1}{2}}q^{\frac{1}{2}}}{ \alpha\gamma^{-\frac{1}{2}}\delta^{-\frac{1}{2}}q^{\frac{1}{2}}}{ \beta\gamma^{-\frac{1}{2}}\delta^{\frac{1}{2}}q^{\frac{1}{2}}}{ \gamma^{\frac{1}{2}}\delta^{-\frac{1}{2}}q^{\frac{1}{2}}}{q}}{\qPochhammer{\alpha q,\beta\delta q,\gamma q}{q}{n}} }

Substitution(s): ν ⁢ ( x ) = 1 2 ⁢ γ 1 2 ⁢ δ 1 2 ⁢ q x + 1 2 + 1 2 ⁢ γ - 1 2 ⁢ δ - 1 2 ⁢ q - x - 1 2 𝜈 𝑥 1 2 superscript 𝛾 1 2 superscript 𝛿 1 2 superscript 𝑞 𝑥 1 2 1 2 superscript 𝛾 1 2 superscript 𝛿 1 2 superscript 𝑞 𝑥 1 2 {\displaystyle{\displaystyle{\displaystyle\nu(x)=\textstyle\frac{1}{2}\gamma^{% \frac{1}{2}}\delta^{\frac{1}{2}}q^{x+\frac{1}{2}}+\textstyle\frac{1}{2}\gamma^% {-\frac{1}{2}}\delta^{-\frac{1}{2}}q^{-x-\frac{1}{2}}}}}


p ~ n ⁡ ( x ; a , b , c , d | q - 1 ) = p ~ n ⁡ ( x ; a - 1 , b - 1 , c - 1 , d - 1 | q ) Askey-Wilson-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 1 Askey-Wilson-polynomial-normalized-p-tilde 𝑛 𝑥 superscript 𝑎 1 superscript 𝑏 1 superscript 𝑐 1 superscript 𝑑 1 𝑞 {\displaystyle{\displaystyle{\displaystyle{\tilde{p}}_{n}\!\left(x;a,b,c,d\,|% \,q^{-1}\right)={\tilde{p}}_{n}\!\left(x;a^{-1},b^{-1},c^{-1},d^{-1}\,|\,q% \right)}}} {\displaystyle \normAskeyWilsonptilde{n}@{x}{a}{b}{c}{d}{q^{-1}}=\normAskeyWilsonptilde{n}@{x}{a^{-1}}{b^{-1}}{c^{-1}}{d^{-1}}{q} }

Koornwinder Addendum: Askey-Wilson

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