# Formula:KLS:14.17:04

$\displaystyle {\displaystyle \sum_{x=0}^N\frac{\qPochhammer{cq^{-N},q^{-N}}{q}{x}}{\qPochhammer{q,cq}{q}{x}} \frac{(1-cq^{2x-N})}{(1-cq^{-N})}c^{-x}q^{x(2N-x)}\dualqKrawtchouk{m}@@{\lambda(x)}{c}{N}{q}\dualqKrawtchouk{n}@@{\lambda(x)}{c}{N}{q} {}=\qPochhammer{c^{-1}}{q}{N}\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q^{-N}}{q}{n}}(cq^{-N})^n\,\Kronecker{m}{n} }$

## Constraint(s)

$\displaystyle {\displaystyle c<0}$

## Substitution(s)

$\displaystyle {\displaystyle \lambda(n)=q^{-n}-pq^n}$ &

$\displaystyle {\displaystyle \lambda(x)=q^{-x}+cq^{x-N}}$ &
$\displaystyle {\displaystyle \lambda(x):=q^{-x}+cq^{x-N}}$ &
$\displaystyle {\displaystyle \lambda(n)=q^{-n}-pq^n}$ &

$\displaystyle {\displaystyle \lambda(x)=q^{-x}+cq^{x-N}}$

## 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

& : logical and
$\displaystyle {\displaystyle \Sigma}$  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
$\displaystyle {\displaystyle (a;q)_n}$  : $\displaystyle {\displaystyle q}$ -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
$\displaystyle {\displaystyle K_{n}}$  : dual $\displaystyle {\displaystyle q}$ -Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk
$\displaystyle {\displaystyle \delta_{m,n}}$  : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4