â« 0 â x c - 1 ⢠( - a ⢠x , - b ⢠q / x ; q ) â ( - x , - q / x ; q ) â ⢠ð x = Ï sin â¡ ( Ï â¢ c ) ⢠( a ⢠b , q c , q 1 - c ; q ) â ( b ⢠q c , a ⢠q - c , q ; q ) â superscript subscript 0 superscript ð¥ ð 1 q-Pochhammer-symbol ð ð¥ ð ð ð¥ ð q-Pochhammer-symbol ð¥ ð ð¥ ð differential-d ð¥ ð q-Pochhammer-symbol ð ð superscript ð ð superscript ð 1 ð ð q-Pochhammer-symbol ð superscript ð ð ð superscript ð ð ð ð {\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}} } â« 0 â x c - 1 ⢠( - a ⢠x ; q ) â ( - x ; q ) â ⢠ð x = Ï sin â¡ ( Ï â¢ c ) ⢠( a , q 1 - c ; q ) â ( a ⢠q - c , q ; q ) â superscript subscript 0 superscript ð¥ ð 1 q-Pochhammer-symbol ð ð¥ ð q-Pochhammer-symbol ð¥ ð differential-d ð¥ ð q-Pochhammer-symbol ð superscript ð 1 ð ð q-Pochhammer-symbol ð superscript ð ð ð ð {\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}} } â« 0 â ( - a ⢠x , - b ⢠q / x ; q ) â ( - x , - q / x ; q ) â ⢠ð x = - ln â¡ q ⢠( a ⢠b , q ; q ) â ( b ⢠q , a / q ; q ) â superscript subscript 0 q-Pochhammer-symbol ð ð¥ ð ð ð¥ ð q-Pochhammer-symbol ð¥ ð ð¥ ð differential-d ð¥ ð q-Pochhammer-symbol ð ð ð ð q-Pochhammer-symbol ð ð ð ð ð {\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 â¢ Ï â¢ â« - 1 1 w ⢠( x ) 1 - x 2 ⢠ð x = 1 2 â¢ Ï â¢ â« 0 Ï w ⢠( cos ⡠θ ) ⢠ð θ = ( a ⢠b ⢠c ⢠d ; q ) â ( a ⢠b , a ⢠c , a ⢠d , b ⢠c , b ⢠d , c ⢠d , q ; q ) â 1 2 superscript subscript 1 1 ð¤ ð¥ 1 superscript ð¥ 2 differential-d ð¥ 1 2 superscript subscript 0 ð¤ ð differential-d ð q-Pochhammer-symbol ð ð ð ð ð q-Pochhammer-symbol ð ð ð ð ð ð ð ð ð ð ð ð ð ð {\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 ) = â k = 0 â ( 1 - 2 ⢠α ⢠x ⢠q k + α 2 ⢠q 2 ⢠k ) = | ( α ⢠e i ⢠θ ; q ) â | 2 = ( α ⢠e i ⢠θ , α ⢠e - i ⢠θ ; q ) â â ð¥ ð¼ superscript subscript product ð 0 1 2 ð¼ ð¥ superscript ð ð superscript ð¼ 2 superscript ð 2 ð superscript q-Pochhammer-symbol ð¼ imaginary-unit ð ð 2 q-Pochhammer-symbol ð¼ imaginary-unit ð ð¼ imaginary-unit ð ð {\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}}}} &
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