# Formula:KLS:14.21:30

$\displaystyle {\displaystyle \int_0^\infty \qLaguerre[\alpha]{m}@{x}{q} \qLaguerre[\alpha]{n}@{x}{q} \frac{x^\alpha}{\qPochhammer{-x}{q}{\infty}} dx=h_n \Kronecker{m}{n} \qquad(\alpha>-1) }$

## Substitution(s)

$\displaystyle {\displaystyle h_n=q^{-\frac12\alpha(\alpha+1)} \qPochhammer{q}{q}{\alpha} \log(q^{-1})\qquad(\alpha\in\ZZ_{\ge0})}$

## Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

## Symbols List

$\displaystyle {\displaystyle \int}$  : integral : http://dlmf.nist.gov/1.4#iv
$\displaystyle {\displaystyle L_n^{(\alpha)}}$  : $\displaystyle {\displaystyle q}$ -Laguerre polynomial : http://drmf.wmflabs.org/wiki/Definition:qLaguerre
$\displaystyle {\displaystyle (a;q)_n}$  : $\displaystyle {\displaystyle q}$ -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
$\displaystyle {\displaystyle \delta_{m,n}}$  : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4
$\displaystyle {\displaystyle \mathrm{log}}$  : principle branch of logarithm logarithm : http://dlmf.nist.gov/4.2#E2