Formula:KLS:14.04:02

<< Formula:KLS:14.04:01

formula in Continuous q-Hahn

Formula:KLS:14.04:07 >>

${\displaystyle{\displaystyle{\displaystyle\frac{1}{4\pi}\int_{-\pi}^{\pi}w(\cos\left(\theta+\phi\right))p_{m}\!\left(\cos\left(\theta+\phi\right);a,b,c,d;q\right)p_{n}\!\left(\cos\left(\theta+\phi\right);a,b,c,d;q\right)\,d\theta{}=\frac{\left(abcdq^{n-1};q\right)_{n}\left(abcdq^{2n};q\right)_{\infty}}{\left(q^{n+1},abq^{n}{\mathrm{e}^{2\mathrm{i}\phi}},acq^{n},adq^{n},bcq^{n},bdq^{n},cdq^{n}{\mathrm{e}^{-2\mathrm{i}\phi}};q\right)_{\infty}}\,\delta_{m,n}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c,d;q)=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}(\theta+\phi)}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm{i}(\theta+2\phi)}},b{\mathrm{e}^{\mathrm{i}(\theta+2\phi)}}c{\mathrm{e}^{\mathrm{i}\theta}},d{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a{\mathrm{e}^{\mathrm{i}\phi}})h(x,b{\mathrm{e}^{\mathrm{i}\phi}})h(x,c{\mathrm{e}^{-\mathrm{i}\phi}})h(x,d{\mathrm{e}^{-\mathrm{i}\phi}})}}}}$ &

${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{\mathrm{i}(\theta+\phi)}},\alpha{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}};q\right)_{\infty}}}}$ &

${\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}$

Proof

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Symbols List

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

Bibliography

Equation in Section 14.4 of KLS.

URL links

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<< Formula:KLS:14.04:01

formula in Continuous q-Hahn

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