# Quantum q-Krawtchouk

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## Basic hypergeometric representation

$\displaystyle {\displaystyle \qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}= \qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{q^{-N}}{q}{pq^{n+1}} }$

Constraint(s): ${\displaystyle {\displaystyle n=0,1,2,\ldots ,N}}$

## Orthogonality relation(s)

$\displaystyle {\displaystyle \sum_{x=0}^N\frac{\qPochhammer{pq}{q}{N-x}}{\qPochhammer{q}{q}{x}\qPochhammer{q}{q}{N-x}}(-1)^{N-x}q^{\binomial{x}{2}} \qtmqKrawtchouk{m}@{q^{-x}}{p}{N}{q}\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q} {}=\frac{(-1)^np^N\qPochhammer{q}{q}{N-n}\qPochhammer{q,pq}{q}{n}}{\qPochhammer{q,q}{q}{N}} q^{\binomial{N+1}{2}-\binomial{n+1}{2}+Nn}\,\Kronecker{m}{n} }$

Constraint(s): ${\displaystyle {\displaystyle p>q^{-N}}}$

## Recurrence relation

$\displaystyle {\displaystyle -pq^{2n+1}(1-q^{-x})\qtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q} {}=(1-q^{n-N})\qtmqKrawtchouk{n+1}@@{q^{-x}}{p}{N}{q} {}-\left[(1-q^{n-N})+q(1-q^n)(1-pq^n)\right]\qtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q} {}+q(1-q^n)(1-pq^n)\qtmqKrawtchouk{n-1}@@{q^{-x}}{p}{N}{q} }$
$\displaystyle {\displaystyle \qtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q}:=\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q} }$

## Monic recurrence relation

$\displaystyle {\displaystyle x\monicqtmqKrawtchouk{n}@@{x}{p}{N}{q}=\monicqtmqKrawtchouk{n+1}@@{x}{p}{N}{q}+ \left[1-p^{-1}q^{-2n-1}\left\{(1-q^{n-N})+q(1-q^n)(1-pq^n)\right\}\right]\monicqtmqKrawtchouk{n}@@{x}{p}{N}{q} {}+p^{-2}q^{-4n+1}(1-q^n)(1-pq^n)(1-q^{n-N-1})\monicqtmqKrawtchouk{n-1}@@{x}{p}{N}{q} }$
$\displaystyle {\displaystyle \qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\frac{p^nq^{n^2}}{\qPochhammer{q^{-N}}{q}{n}}\monicqtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q} }$

## q-Difference equation

${\displaystyle {\displaystyle -p(1-q^{n})y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1)}}$

Substitution(s): ${\displaystyle {\displaystyle D(x)=(1-q^{x})(p-q^{x-N-1})}}$ &

${\displaystyle {\displaystyle B(x)=-q^{x}(1-q^{x-N})}}$ &

$\displaystyle {\displaystyle y(x)=\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}}$

## Forward shift operator

$\displaystyle {\displaystyle \qtmqKrawtchouk{n}@{q^{-x-1}}{p}{N}{q}-\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q} {}=\frac{pq^{-x}(1-q^n)}{1-q^{-N}}\qtmqKrawtchouk{n-1}@{q^{-x}}{pq}{N-1}{q} }$
$\displaystyle {\displaystyle \frac{\Delta \qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}}{\Delta q^{-x}}= \frac{pq(1-q^n)}{(1-q)(1-q^{-N})}\qtmqKrawtchouk{n-1}@{q^{-x}}{pq}{N-1}{q} }$

## Backward shift operator

$\displaystyle {\displaystyle (1-q^{x-N-1})\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q} {}+q^{-x}(1-q^x)(p-q^{x-N-1})\qtmqKrawtchouk{n}@{q^{-x+1}}{p}{N}{q} {}=(1-q^{-N-1})\qtmqKrawtchouk{n+1}@{q^{-x}}{pq^{-1}}{N+1}{q} }$
$\displaystyle {\displaystyle \frac{\nabla\left[w(x;p,N;q)\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}\right]}{\nabla q^{-x}} {}=\frac{1}{1-q}w(x;pq^{-1},N+1;q)\qtmqKrawtchouk{n+1}@{q^{-x}}{pq^{-1}}{N+1}{q} }$

Substitution(s): $\displaystyle {\displaystyle w(x;p,N;q)=\frac{\qPochhammer{q^{-N}}{q}{x}}{\qPochhammer{q,p^{-1}q^{-N}}{q}{x}}(-p)^{-x}q^{\binomial{x+1}{2}}}$

## Rodrigues-type formula

$\displaystyle {\displaystyle w(x;p,N;q)\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=(1-q)^n\left(\nabla_q\right)^n\left[w(x;pq^n,N-n;q)\right] }$

Substitution(s): $\displaystyle {\displaystyle w(x;p,N;q)=\frac{\qPochhammer{q^{-N}}{q}{x}}{\qPochhammer{q,p^{-1}q^{-N}}{q}{x}}(-p)^{-x}q^{\binomial{x+1}{2}}}$

${\displaystyle {\displaystyle \nabla _{q}:={\frac {\nabla }{\nabla q^{-x}}}}}$

## Generating functions

$\displaystyle {\displaystyle \qPochhammer{q^{x-N}t}{q}{N-x}\cdot\qHyperrphis{2}{1}@@{q^{-x},pq^{N+1-x}}{0}{q}{q^{x-N}t} {}=\sum_{n=0}^N\frac{\qPochhammer{q^{-N}}{q}{n}}{\qPochhammer{q}{q}{n}}\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}t^n }$
$\displaystyle {\displaystyle \qPochhammer{q^{-x}t}{q}{x}\cdot\qHyperrphis{2}{1}@@{q^{x-N},0}{pq}{q}{q^{-x}t} {}=\sum_{n=0}^N\frac{\qPochhammer{q^{-N}}{q}{n}}{\qPochhammer{pq,q}{q}{n}}\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}t^n }$

## Limit relations

### q-Hahn polynomial to Quantum q-Krawtchouk polynomial

$\displaystyle {\displaystyle \lim_{\alpha\rightarrow\infty}\qHahn{n}@{q^{-x}}{\alpha}{p}{N}{q}=\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q} }$

### Quantum q-Krawtchouk polynomial to Al-Salam-Carlitz II polynomial

$\displaystyle {\displaystyle \lim_{N\rightarrow\infty}\qtmqKrawtchouk{n}@{x}{a^{-1}q^{-N-1}}{N}{q}= \left(-\frac{1}{a}\right)^nq^{\binomial{n}{2}}\AlSalamCarlitzII{a}{n}@{x}{q} }$

### Quantum q-Krawtchouk polynomial to Krawtchouk polynomial

$\displaystyle {\displaystyle \lim_{q\rightarrow 1}\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\Krawtchouk{n}@{x}{p^{-1}}{N} }$

## Remarks

$\displaystyle {\displaystyle \qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\qMeixner{n}@{q^{-x}}{q^{-N-1}}{-p^{-1}}{q} }$
$\displaystyle {\displaystyle \qtmqKrawtchouk{n}@{q^x}{p}{N}{q^{-1}}=\qPochhammer{p^{-1}q}{q}{n}\left(-\frac{p}{q}\right)^nq^{-\binomial{n}{2}} \AffqKrawtchouk{n}@{q^{x-N}}{p^{-1}}{N}{q} }$