# Formula:KLS:14.11:03

$\displaystyle {\displaystyle \int_{bq}^{aq}\frac{\qPochhammer{a^{-1}x,b^{-1}x}{q}{\infty}}{\qPochhammer{x}{q}{\infty}} \bigqLaguerre{m}@{x}{a}{b}{q}\bigqLaguerre{n}@{x}{a}{b}{q}\,d_qx {}=aq(1-q)\frac{\qPochhammer{q,a^{-1}b,ab^{-1}q}{q}{\infty}} {\qPochhammer{aq,bq}{q}{\infty}}\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{aq,bq}{q}{n}}(-abq^2)^nq^{\binomial{n}{2}}\,\Kronecker{m}{n} }$

## Proof

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## Symbols List

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