Formula:KLS:14.19:03

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

Substitution(s)

w ⁢ ( x ) := w ⁢ ( x ; q α | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ ⁢ q 1 2 ⁢ α + 3 4 ⁢ e i ⁢ θ ; q ) ∞ | 2 = | ( e i ⁢ θ , - e i ⁢ θ ; q 1 2 ) ∞ ( q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ ; q 1 2 ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , q 1 2 ⁢ α + 1 4 ) ⁢ h ⁢ ( x , q 1 2 ⁢ α + 3 4 ) assign 𝑤 𝑥 𝑤 𝑥 conditional superscript 𝑞 𝛼 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛼 3 4 imaginary-unit 𝜃 𝑞 2 superscript q-Pochhammer-symbol imaginary-unit 𝜃 imaginary-unit 𝜃 superscript 𝑞 1 2 q-Pochhammer-symbol superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 𝛼 1 4 ℎ 𝑥 superscript 𝑞 1 2 𝛼 3 4 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha}|q)=\left|\frac% {\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(q^{\frac{1}{2% }\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}q^{\frac{1}{2}\alpha+\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(q^{\frac{1}{2}\alpha+\frac{1}{4}}{% \mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\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,q^{\frac{1}{% 2}\alpha+\frac{1}{4}})h(x,q^{\frac{1}{2}\alpha+\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}}}


Proof

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Symbols List

& : logical and
∫ {\displaystyle{\displaystyle{\displaystyle\int}}}  : integral : http://dlmf.nist.gov/1.4#iv
P α ( n ) subscript superscript 𝑃 𝑛 𝛼 {\displaystyle{\displaystyle{\displaystyle P^{(n)}_{\alpha}}}}  : continuous q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Laguerre polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre
( a ; q ) n subscript 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
δ m , n subscript 𝛿 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}  : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4
e e {\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i
Π Π {\displaystyle{\displaystyle{\displaystyle\Pi}}}  : product : http://drmf.wmflabs.org/wiki/Definition:prod
cos cos {\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}  : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.19 of KLS.

URL links

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