Formula:KLS:14.03:02

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1 2 ⁢ π ⁢ ∫ - 1 1 w ⁢ ( x ) 1 - x 2 ⁢ p m ⁡ ( x ; a , b , c | q ) ⁢ p n ⁡ ( x ; a , b , c | q ) ⁢ 𝑑 x = h n ⁢ δ m , n 1 2 superscript subscript 1 1 𝑤 𝑥 1 superscript 𝑥 2 continuous-dual-q-Hahn-polynomial-p 𝑚 𝑥 𝑎 𝑏 𝑐 𝑞 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 differential-d 𝑥 subscript ℎ 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x% )}{\sqrt{1-x^{2}}}p_{m}\!\left(x;a,b,c|q\right)p_{n}\!\left(x;a,b,c|q\right)\,% dx=h_{n}\,\delta_{m,n}}}}

Substitution(s)

h n = 1 ( q n + 1 , a ⁢ b ⁢ q n , a ⁢ c ⁢ q n , b ⁢ c ⁢ q n ; q ) ∞ subscript ℎ 𝑛 1 q-Pochhammer-symbol superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 𝑎 𝑐 superscript 𝑞 𝑛 𝑏 𝑐 superscript 𝑞 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle h_{n}=\frac{1}{\left(q^{n+1},abq^{n% },acq^{n},bcq^{n};q\right)_{\infty}}}}} &

w ⁢ ( x ) := w ⁢ ( x ; a , b , c | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ , c ⁢ e i ⁢ θ ; q ) ∞ | 2 = 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 ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑐 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 𝑎 ℎ 𝑥 𝑏 ℎ 𝑥 𝑐 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c|q)=\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}};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,a)h(x,b)h(x,c)}}}} &
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 n subscript 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle p_{n}}}}  : continuous dual q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn
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
( 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
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 14.3 of KLS.

URL links

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