∫ 0 ∞ x c - 1 ( - a x , - b q / x ; q ) ∞ ( - x , - q / x ; q ) ∞ 𝑑 x = π sin ( π c ) ( a b , q c , q 1 - c ; q ) ∞ ( b q c , a q - c , q ; q ) ∞ superscript subscript 0 superscript 𝑥 𝑐 1 q-Pochhammer-symbol 𝑎 𝑥 𝑏 𝑞 𝑥 𝑞 q-Pochhammer-symbol 𝑥 𝑞 𝑥 𝑞 differential-d 𝑥 𝑐 q-Pochhammer-symbol 𝑎 𝑏 superscript 𝑞 𝑐 superscript 𝑞 1 𝑐 𝑞 q-Pochhammer-symbol 𝑏 superscript 𝑞 𝑐 𝑎 superscript 𝑞 𝑐 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle\int_{0}^{\infty}x^{c-1}\frac{\left(% -ax,-bq/x;q\right)_{\infty}}{\left(-x,-q/x;q\right)_{\infty}}\,dx=\frac{\pi}{% \sin\left(\pi c\right)}\,\frac{\left(ab,q^{c},q^{1-c};q\right)_{\infty}}{\left% (bq^{c},aq^{-c},q;q\right)_{\infty}}}}} {\displaystyle \int_0^{\infty}x^{c-1}\frac{\qPochhammer{-ax,-bq/x}{q}{\infty}}{\qPochhammer{-x,-q/x}{q}{\infty}}\,dx =\frac{\cpi}{\sin@{\cpi c}}\,\frac{\qPochhammer{ab,q^c,q^{1-c}}{q}{\infty}}{\qPochhammer{bq^c,aq^{-c},q}{q}{\infty}} } ∫ 0 ∞ x c - 1 ( - a x ; q ) ∞ ( - x ; q ) ∞ 𝑑 x = π sin ( π c ) ( a , q 1 - c ; q ) ∞ ( a q - c , q ; q ) ∞ superscript subscript 0 superscript 𝑥 𝑐 1 q-Pochhammer-symbol 𝑎 𝑥 𝑞 q-Pochhammer-symbol 𝑥 𝑞 differential-d 𝑥 𝑐 q-Pochhammer-symbol 𝑎 superscript 𝑞 1 𝑐 𝑞 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑐 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle\int_{0}^{\infty}x^{c-1}\frac{\left(% -ax;q\right)_{\infty}}{\left(-x;q\right)_{\infty}}\,dx=\frac{\pi}{\sin\left(% \pi c\right)}\,\frac{\left(a,q^{1-c};q\right)_{\infty}}{\left(aq^{-c},q;q% \right)_{\infty}}}}} {\displaystyle \int_0^{\infty}x^{c-1}\frac{\qPochhammer{-ax}{q}{\infty}}{\qPochhammer{-x}{q}{\infty}}\,dx =\frac{\cpi}{\sin@{\cpi c}}\,\frac{\qPochhammer{a,q^{1-c}}{q}{\infty}}{\qPochhammer{aq^{-c},q}{q}{\infty}} } ∫ 0 ∞ ( - a x , - b q / x ; q ) ∞ ( - x , - q / x ; q ) ∞ 𝑑 x = - ln q ( a b , q ; q ) ∞ ( b q , a / q ; q ) ∞ superscript subscript 0 q-Pochhammer-symbol 𝑎 𝑥 𝑏 𝑞 𝑥 𝑞 q-Pochhammer-symbol 𝑥 𝑞 𝑥 𝑞 differential-d 𝑥 𝑞 q-Pochhammer-symbol 𝑎 𝑏 𝑞 𝑞 q-Pochhammer-symbol 𝑏 𝑞 𝑎 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle\int_{0}^{\infty}\frac{\left(-ax,-bq% /x;q\right)_{\infty}}{\left(-x,-q/x;q\right)_{\infty}}\,dx=-\ln q\,\frac{\left% (ab,q;q\right)_{\infty}}{\left(bq,a/q;q\right)_{\infty}}}}} {\displaystyle \int_0^{\infty}\frac{\qPochhammer{-ax,-bq/x}{q}{\infty}}{\qPochhammer{-x,-q/x}{q}{\infty}}\,dx =-\ln@@{q}\,\frac{\qPochhammer{ab,q}{q}{\infty}}{\qPochhammer{bq,a/q}{q}{\infty}} } 1 2 π ∫ - 1 1 w ( x ) 1 - x 2 𝑑 x = 1 2 π ∫ 0 π w ( cos θ ) 𝑑 θ = ( a b c d ; q ) ∞ ( a b , a c , a d , b c , b d , c d , q ; q ) ∞ 1 2 superscript subscript 1 1 𝑤 𝑥 1 superscript 𝑥 2 differential-d 𝑥 1 2 superscript subscript 0 𝑤 𝜃 differential-d 𝜃 q-Pochhammer-symbol 𝑎 𝑏 𝑐 𝑑 𝑞 q-Pochhammer-symbol 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 𝑏 𝑐 𝑏 𝑑 𝑐 𝑑 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle{}{}{}\frac{1}{2\pi}\int_{-1}^{1}% \frac{w(x)}{\sqrt{1-x^{2}}}\,dx=\frac{1}{2\pi}\int_{0}^{\pi}w(\cos\theta)\,d% \theta=\frac{\left(abcd;q\right)_{\infty}}{\left(ab,ac,ad,bc,bd,cd,q;q\right)_% {\infty}}}}} {\displaystyle \index{Askey-Wilson integral}\index{Askey-Wilson q-beta integral@Askey-Wilson $q$-beta integral}\index{q-Beta integral@$q$-Beta integral} \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\,dx=\frac{1}{2\cpi}\int_0^{\cpi}w(\cos@@{\theta})\,d\theta =\frac{\qPochhammer{abcd}{q}{\infty}}{\qPochhammer{ab,ac,ad,bc,bd,cd,q}{q}{\infty}} }
h ( x , α ) = ∏ k = 0 ∞ ( 1 - 2 α x q k + α 2 q 2 k ) = | ( α e i θ ; q ) ∞ | 2 = ( α e i θ , α e - i θ ; q ) ∞ ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 superscript q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 𝑞 2 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|\left(\alpha{\mathrm{e}^{% \mathrm{i}\theta}};q\right)_{\infty}\right|^{2}=\left(\alpha{\mathrm{e}^{% \mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &