Affine q-Krawtchouk

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Affine q-Krawtchouk

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Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)=\qHyperrphis{3}{2}@@{q^{-n},0,q^{-x}}{pq,q^{-N}}{q}{q}}}}$

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

${\displaystyle{\displaystyle{\displaystyle K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)=\frac{(-pq)^{n}q^{\binomial{n}{2}}}{\left(pq;q\right)_{n}}\ \qHyperrphis{2}{1}@@{q^{-n},q^{x-N}}{q^{-N}}{q}{\frac{q^{-x}}{p}}}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\frac{\left(pq;q\right)_{x}\left(q;q\right)_{N}}{\left(q;q\right)_{x}\left(q;q\right)_{N-x}}(pq)^{-x}K^{\mathrm{Aff}}_{m}\!\left(q^{-x};p,N;q\right)K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right){}=(pq)^{n-N}\frac{\left(q;q\right)_{n}\left(q;q\right)_{N-n}}{\left(pq;q\right)_{n}\left(q;q\right)_{N}}\,\delta_{m,n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle 0<pq<1}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle-(1-q^{-x})K^{\mathrm{Aff}}_{n}\!\left(q^{-x}\right){}=(1-q^{n-N})(1-pq^{n+1})K^{\mathrm{Aff}}_{n+1}\!\left(q^{-x}\right){}-\left[(1-q^{n-N})(1-pq^{n+1})-pq^{n-N}(1-q^{n})\right]K^{\mathrm{Aff}}_{n}\!\left(q^{-x}\right){}-pq^{n-N}(1-q^{n})K^{\mathrm{Aff}}_{n-1}\!\left(q^{-x}\right)}}}$
${\displaystyle{\displaystyle{\displaystyle K^{\mathrm{Aff}}_{n}\!\left(q^{-x}\right):=K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x\widehat{K}^{\mathrm{Aff}}_{n}\!\left(x\right)=\widehat{K}^{\mathrm{Aff}}_{n+1}\!\left(x\right)+\left[1-\left\{(1-q^{n-N})(1-pq^{n+1})-pq^{n-N}(1-q^{n})\right\}\right]\widehat{K}^{\mathrm{Aff}}_{n}\!\left(x\right){}-pq^{n-N}(1-q^{n})(1-pq^{n})(1-q^{n-N-1})\widehat{K}^{\mathrm{Aff}}_{n-1}\!\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)=\frac{1}{\left(pq,q^{-N};q\right)_{n}}\widehat{K}^{\mathrm{Aff}}_{n}\!\left(q^{-x}\right)}}}$

q-Difference equation

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

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

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

${\displaystyle{\displaystyle{\displaystyle y(x)=K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle K^{\mathrm{Aff}}_{n}\!\left(q^{-x-1};p,N;q\right)-K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right){}=\frac{q^{-n-x}(1-q^{n})}{(1-pq)(1-q^{-N})}K^{\mathrm{Aff}}_{n-1}\!\left(q^{-x};pq,N-1;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{\Delta K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)}{\Delta q^{-x}}{}=\frac{q^{-n+1}(1-q^{n})}{(1-q)(1-pq)(1-q^{-N})}K^{\mathrm{Aff}}_{n-1}\!\left(q^{-x};pq,N-1;q\right)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle(1-pq^{x})(1-q^{-x+N+1})K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)-p(1-q^{x})K^{\mathrm{Aff}}_{n}\!\left(q^{-x+1};p,N;q\right){}=(1-p)(1-q^{N+1})K^{\mathrm{Aff}}_{n+1}\!\left(q^{-x};pq^{-1},N+1;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{\nabla\left[w(x;p,N;q)K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)\right]}{\nabla q^{-x}}{}=\frac{1-q^{N+1}}{1-q}w(x;pq^{-1},N+1;q)K^{\mathrm{Aff}}_{n+1}\!\left(q^{-x};pq^{-1},N+1;q\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;p,N;q)=\frac{\left(pq;q\right)_{x}}{\left(q;q\right)_{x}\left(q;q\right)_{N-x}}p^{-x}}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle w(x;p,N;q)K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right){}=\frac{(-1)^{n}q^{-Nn+\binomial{n}{2}}(1-q)^{n}}{\left(q^{-N};q\right)_{n}}\left(\nabla_{q}\right)^{n}\left[w(x;pq^{n},N-n;q)\right]}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;p,N;q)=\frac{\left(pq;q\right)_{x}}{\left(q;q\right)_{x}\left(q;q\right)_{N-x}}p^{-x}}}}$

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

Generating functions

${\displaystyle{\displaystyle{\displaystyle\left(q^{-N}t;q\right)_{N-x}\cdot\qHyperrphis{1}{1}@@{q^{-x}}{pq}{q}{pqt}=\sum_{n=0}^{N}\frac{\left(q^{-N};q\right)_{n}}{\left(q;q\right)_{n}}K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\left(-pq^{-N+1}t;q\right)_{x}\cdot\qHyperrphis{2}{0}@@{q^{x-N},pq^{x+1}}{-}{q}{-q^{-x}t}{}=\sum_{n=0}^{N}\frac{\left(pq,q^{-N};q\right)_{n}}{\left(q;q\right)_{n}}q^{-\binomial{n}{2}}K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)t^{n}}}}$

Limit relations

q-Hahn polynomial to Affine q-Krawtchouk polynomial

${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(q^{-x};p,0,N;q\right)=K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)}}}$

Dual q-Hahn polynomial to Affine q-Krawtchouk polynomial

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);p,0,N\right){q}=K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)}}}$

Affine q-Krawtchouk polynomial to Little q-Laguerre / Wall polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}K^{\mathrm{Aff}}_{n}\!\left(q^{x-N};p,N;q\right)=p_{n}\!\left(q^{x};p;q\right)}}}$

Affine q-Krawtchouk polynomial to Krawtchouk polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)=K_{n}\!\left(x;1-p,N\right)}}}$

Remarks

${\displaystyle{\displaystyle{\displaystyle K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)=P_{n}\!\left(q^{-x};p,q^{-N-1};q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle K^{\mathrm{Aff}}_{n}\!\left(q^{x};p,N;q^{-1}\right)=\frac{1}{\left(p^{-1}q;q\right)_{n}}K^{\mathrm{qtm}}_{n}\!\left(q^{x-N};p^{-1},N;q\right)}}}$

Koornwinder Addendum: Affine q-Krawtchouk

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