# Formula:KLS:14.28:04

$\displaystyle {\displaystyle \int_{-1}^1\qPochhammer{qx,-qx}{q}{\infty}\discrqHermiteI{m}@{x}{q}\discrqHermiteI{n}@{x}{q}\,d_qx {}=(1-q)\qPochhammer{q}{q}{n}\qPochhammer{q,-1,-q}{q}{\infty}q^{\binomial{n}{2}}\,\Kronecker{m}{n} }$

## 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
$\displaystyle {\displaystyle h_{n}}$  : discrete $\displaystyle {\displaystyle q}$ -Hermite I polynomial : http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI
$\displaystyle {\displaystyle \binom{n}{k}}$  : 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