Formula:KLS:14.05:02

<< Formula:KLS:14.05:01

formula in Big q-Jacobi

Formula:KLS:14.05:03 >>

${\displaystyle{\displaystyle{\displaystyle\int_{cq}^{aq}\frac{\left(a^{-1}x,c^{-1}x;q\right)_{\infty}}{\left(x,bc^{-1}x;q\right)_{\infty}}P_{m}\!\left(x;a,b,c;q\right)P_{n}\!\left(x;a,b,c;q\right)\,d_{q}x{}=aq(1-q)\frac{\left(q,abq^{2},a^{-1}c,ac^{-1}q;q\right)_{\infty}}{\left(aq,bq,cq,abc^{-1}q;q\right)_{\infty}}{}\frac{(1-abq)}{(1-abq^{2n+1})}\frac{\left(q,bq,abc^{-1}q;q\right)_{n}}{\left(aq,abq,cq;q\right)_{n}}(-acq^{2})^{n}q^{\binomial{n}{2}}\,\delta_{m,n}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : big ${\displaystyle{\displaystyle{\displaystyle q}}}$-Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:bigqJacobi
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.5 of KLS.

URL links

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formula in Big q-Jacobi

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