More integrals

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More integrals

∫ 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}} }

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

x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


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