Formula:KLS:14.10:38: Difference between revisions

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Latest revision as of 08:37, 22 December 2019


1 2 ⁢ π ⁢ ∫ - 1 1 w ⁢ ( x ) 1 - x 2 ⁢ C m ⁡ ( x ; β | q ) ⁢ C n ⁡ ( x ; β | q ) ⁢ 𝑑 x = ( β , β ⁢ q ; q ) ∞ ( β 2 , q ; q ) ∞ ⁢ ( β 2 ; q ) n ( q ; q ) n ⁢ ( 1 - β ) ( 1 - β ⁢ q n ) ⁢ δ m , n 1 2 superscript subscript 1 1 𝑤 𝑥 1 superscript 𝑥 2 continuous-q-ultraspherical-Rogers-polynomial 𝑚 𝑥 𝛽 𝑞 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 differential-d 𝑥 q-Pochhammer-symbol 𝛽 𝛽 𝑞 𝑞 q-Pochhammer-symbol superscript 𝛽 2 𝑞 𝑞 q-Pochhammer-symbol superscript 𝛽 2 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 1 𝛽 1 𝛽 superscript 𝑞 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x% )}{\sqrt{1-x^{2}}}C_{m}\!\left(x;\beta\,|\,q\right)C_{n}\!\left(x;\beta\,|\,q% \right)\,dx{}=\frac{\left(\beta,\beta q;q\right)_{\infty}}{\left(\beta^{2},q;q% \right)_{\infty}}\frac{\left(\beta^{2};q\right)_{n}}{\left(q;q\right)_{n}}% \frac{(1-\beta)}{(1-\beta q^{n})}\,\delta_{m,n}}}}

Constraint(s)

| β | < 1 𝛽 1 {\displaystyle{\displaystyle{\displaystyle|\beta|<1}}}


Substitution(s)

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

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
C n subscript 𝐶 𝑛 {\displaystyle{\displaystyle{\displaystyle C_{n}}}}  : continuous q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -ultraspherical/Rogers polynomial : http://dlmf.nist.gov/18.28#E13
( 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 14.10 of KLS.

URL links

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