# Formula:KLS:14.17:19

$\displaystyle {\displaystyle w(x;c,N|q)\dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q} {}=(1-q)^n\qPochhammer{cq^{-N}}{q}{n}\left(\nabla_{\lambda}\right)^n\left[w(x;c,N-n|q)\right] }$

## Substitution(s)

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

$\displaystyle {\displaystyle \lambda(x)=q^{-x}+cq^{x-N}}$ &
$\displaystyle {\displaystyle w(x;c,N|q)=\frac{\qPochhammer{q^{-N},cq^{-N}}{q}{x}}{\qPochhammer{q,cq}{q}{x}}c^{-x}q^{2Nx-x(x-1)}}$ &
$\displaystyle {\displaystyle \lambda(x):=q^{-x}+cq^{x-N}}$ &
$\displaystyle {\displaystyle \lambda(n)=q^{-n}-pq^n}$ &

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

## 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
$\displaystyle {\displaystyle K_{n}}$  : dual $\displaystyle {\displaystyle q}$ -Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk
$\displaystyle {\displaystyle (a;q)_n}$  : $\displaystyle {\displaystyle q}$ -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1