Affine q-Krawtchouk

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Affine q-Krawtchouk

Basic hypergeometric representation

K n Aff ⁡ ( q - x ; p , N ; q ) = \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ q - n , 0 , q - x ⁢ p ⁢ q , q - N ⁢ q ⁢ q affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 0 superscript 𝑞 𝑥 𝑝 𝑞 superscript 𝑞 𝑁 𝑞 𝑞 {\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}}}} {\displaystyle \AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\qHyperrphis{3}{2}@@{q^{-n},0,q^{-x}}{pq,q^{-N}}{q}{q} }

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


K n Aff ⁡ ( q - x ; p , N ; q ) = ( - p ⁢ q ) n ⁢ q \binomial ⁢ n ⁢ 2 ( p ⁢ q ; q ) n ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - n , q x - N ⁢ q - N ⁢ q ⁢ q - x p affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 superscript 𝑝 𝑞 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Pochhammer-symbol 𝑝 𝑞 𝑞 𝑛 \qHyperrphis 21 @ @ superscript 𝑞 𝑛 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑁 𝑞 superscript 𝑞 𝑥 𝑝 {\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}}}}} {\displaystyle \AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\frac{(-pq)^nq^{\binomial{n}{2}}}{\qPochhammer{pq}{q}{n}}\ \qHyperrphis{2}{1}@@{q^{-n},q^{x-N}}{q^{-N}}{q}{\frac{q^{-x}}{p}} }

Orthogonality relation(s)

∑ x = 0 N ( p ⁢ q ; q ) x ⁢ ( q ; q ) N ( q ; q ) x ⁢ ( q ; q ) N - x ⁢ ( p ⁢ q ) - x ⁢ K m Aff ⁡ ( q - x ; p , N ; q ) ⁢ K n Aff ⁡ ( q - x ; p , N ; q ) = ( p ⁢ q ) n - N ⁢ ( q ; q ) n ⁢ ( q ; q ) N - n ( p ⁢ q ; q ) n ⁢ ( q ; q ) N ⁢ δ m , n superscript subscript 𝑥 0 𝑁 q-Pochhammer-symbol 𝑝 𝑞 𝑞 𝑥 q-Pochhammer-symbol 𝑞 𝑞 𝑁 q-Pochhammer-symbol 𝑞 𝑞 𝑥 q-Pochhammer-symbol 𝑞 𝑞 𝑁 𝑥 superscript 𝑝 𝑞 𝑥 affine-q-Krawtchouk-polynomial-K 𝑚 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 superscript 𝑝 𝑞 𝑛 𝑁 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑁 𝑛 q-Pochhammer-symbol 𝑝 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑁 Kronecker-delta 𝑚 𝑛 {\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}}}} {\displaystyle \sum_{x=0}^N\frac{\qPochhammer{pq}{q}{x}\qPochhammer{q}{q}{N}}{\qPochhammer{q}{q}{x}\qPochhammer{q}{q}{N-x}}(pq)^{-x}\AffqKrawtchouk{m}@{q^{-x}}{p}{N}{q}\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q} {}=(pq)^{n-N}\frac{\qPochhammer{q}{q}{n}\qPochhammer{q}{q}{N-n}}{\qPochhammer{pq}{q}{n}\qPochhammer{q}{q}{N}}\,\Kronecker{m}{n} }

Constraint(s): 0 < p ⁢ q < 1 0 𝑝 𝑞 1 {\displaystyle{\displaystyle{\displaystyle 0<pq<1}}}


Recurrence relation

- ( 1 - q - x ) ⁢ K n Aff ⁡ ( q - x ) = ( 1 - q n - N ) ⁢ ( 1 - p ⁢ q n + 1 ) ⁢ K n + 1 Aff ⁡ ( q - x ) - [ ( 1 - q n - N ) ⁢ ( 1 - p ⁢ q n + 1 ) - p ⁢ q n - N ⁢ ( 1 - q n ) ] ⁢ K n Aff ⁡ ( q - x ) - p ⁢ q n - N ⁢ ( 1 - q n ) ⁢ K n - 1 Aff ⁡ ( q - x ) 1 superscript 𝑞 𝑥 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 1 superscript 𝑞 𝑛 𝑁 1 𝑝 superscript 𝑞 𝑛 1 affine-q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 delimited-[] 1 superscript 𝑞 𝑛 𝑁 1 𝑝 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 𝑛 𝑁 1 superscript 𝑞 𝑛 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 𝑝 superscript 𝑞 𝑛 𝑁 1 superscript 𝑞 𝑛 affine-q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\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 -(1-q^{-x})\AffqKrawtchouk{n}@@{q^{-x}}{p}{N}{q} {}=(1-q^{n-N})(1-pq^{n+1})\AffqKrawtchouk{n+1}@@{q^{-x}}{p}{N}{q} {}-\left[(1-q^{n-N})(1-pq^{n+1})-pq^{n-N}(1-q^n)\right]\AffqKrawtchouk{n}@@{q^{-x}}{p}{N}{q} {}-pq^{n-N}(1-q^n)\AffqKrawtchouk{n-1}@@{q^{-x}}{p}{N}{q} }
K n Aff ⁡ ( q - x ) := K n Aff ⁡ ( q - x ; p , N ; q ) assign affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle K^{\mathrm{Aff}}_{n}\!\left(q^{-x}% \right):=K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)}}} {\displaystyle \AffqKrawtchouk{n}@@{q^{-x}}{p}{N}{q}:=\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q} }

Monic recurrence relation

x ⁢ K ^ n Aff ⁡ ( x ) = K ^ n + 1 Aff ⁡ ( x ) + [ 1 - { ( 1 - q n - N ) ⁢ ( 1 - p ⁢ q n + 1 ) - p ⁢ q n - N ⁢ ( 1 - q n ) } ] ⁢ K ^ n Aff ⁡ ( x ) - p ⁢ q n - N ⁢ ( 1 - q n ) ⁢ ( 1 - p ⁢ q n ) ⁢ ( 1 - q n - N - 1 ) ⁢ K ^ n - 1 Aff ⁡ ( x ) 𝑥 affine-q-Krawtchouk-polynomial-monic-p 𝑛 𝑥 𝑝 𝑁 𝑞 affine-q-Krawtchouk-polynomial-monic-p 𝑛 1 𝑥 𝑝 𝑁 𝑞 delimited-[] 1 1 superscript 𝑞 𝑛 𝑁 1 𝑝 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 𝑛 𝑁 1 superscript 𝑞 𝑛 affine-q-Krawtchouk-polynomial-monic-p 𝑛 𝑥 𝑝 𝑁 𝑞 𝑝 superscript 𝑞 𝑛 𝑁 1 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 𝑁 1 affine-q-Krawtchouk-polynomial-monic-p 𝑛 1 𝑥 𝑝 𝑁 𝑞 {\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 x\monicAffqKrawtchouk{n}@@{x}{p}{N}{q}=\monicAffqKrawtchouk{n+1}@@{x}{p}{N}{q}+ \left[1-\left\{(1-q^{n-N})(1-pq^{n+1})-pq^{n-N}(1-q^n)\right\}\right]\monicAffqKrawtchouk{n}@@{x}{p}{N}{q} {}-pq^{n-N}(1-q^n)(1-pq^n)(1-q^{n-N-1})\monicAffqKrawtchouk{n-1}@@{x}{p}{N}{q} }
K n Aff ⁡ ( q - x ; p , N ; q ) = 1 ( p ⁢ q , q - N ; q ) n ⁢ K ^ n Aff ⁡ ( q - x ) affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 1 q-Pochhammer-symbol 𝑝 𝑞 superscript 𝑞 𝑁 𝑞 𝑛 affine-q-Krawtchouk-polynomial-monic-p 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\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)}}} {\displaystyle \AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\frac{1}{\qPochhammer{pq,q^{-N}}{q}{n}}\monicAffqKrawtchouk{n}@@{q^{-x}}{p}{N}{q} }

q-Difference equation

q - n ⁢ ( 1 - q n ) ⁢ y ⁢ ( x ) = B ⁢ ( x ) ⁢ y ⁢ ( x + 1 ) - [ B ⁢ ( x ) + D ⁢ ( x ) ] ⁢ y ⁢ ( x ) + D ⁢ ( x ) ⁢ y ⁢ ( x - 1 ) superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 𝑦 𝑥 𝐵 𝑥 𝑦 𝑥 1 delimited-[] 𝐵 𝑥 𝐷 𝑥 𝑦 𝑥 𝐷 𝑥 𝑦 𝑥 1 {\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)}}} {\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): D ⁢ ( x ) = - p ⁢ ( 1 - q x ) ⁢ q x - N 𝐷 𝑥 𝑝 1 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle D(x)=-p(1-q^{x})q^{x-N}}}} &

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

y ⁢ ( x ) = K n Aff ⁡ ( q - x ; p , N ; q ) 𝑦 𝑥 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=K^{\mathrm{Aff}}_{n}\!\left(q^% {-x};p,N;q\right)}}}


Forward shift operator

K n Aff ⁡ ( q - x - 1 ; p , N ; q ) - K n Aff ⁡ ( q - x ; p , N ; q ) = q - n - x ⁢ ( 1 - q n ) ( 1 - p ⁢ q ) ⁢ ( 1 - q - N ) ⁢ K n - 1 Aff ⁡ ( q - x ; p ⁢ q , N - 1 ; q ) affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 1 𝑝 𝑁 𝑞 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 superscript 𝑞 𝑛 𝑥 1 superscript 𝑞 𝑛 1 𝑝 𝑞 1 superscript 𝑞 𝑁 affine-q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 𝑞 𝑁 1 𝑞 {\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 \AffqKrawtchouk{n}@{q^{-x-1}}{p}{N}{q}-\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q} {}=\frac{q^{-n-x}(1-q^n)}{(1-pq)(1-q^{-N})}\AffqKrawtchouk{n-1}@{q^{-x}}{pq}{N-1}{q} }
Δ ⁢ K n Aff ⁡ ( q - x ; p , N ; q ) Δ ⁢ q - x = q - n + 1 ⁢ ( 1 - q n ) ( 1 - q ) ⁢ ( 1 - p ⁢ q ) ⁢ ( 1 - q - N ) ⁢ K n - 1 Aff ⁡ ( q - x ; p ⁢ q , N - 1 ; q ) Δ affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 Δ superscript 𝑞 𝑥 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 𝑞 1 𝑝 𝑞 1 superscript 𝑞 𝑁 affine-q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 𝑞 𝑁 1 𝑞 {\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)}}} {\displaystyle \frac{\Delta \AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}}{\Delta q^{-x}} {}=\frac{q^{-n+1}(1-q^n)}{(1-q)(1-pq)(1-q^{-N})}\AffqKrawtchouk{n-1}@{q^{-x}}{pq}{N-1}{q} }

Backward shift operator

( 1 - p ⁢ q x ) ⁢ ( 1 - q - x + N + 1 ) ⁢ K n Aff ⁡ ( q - x ; p , N ; q ) - p ⁢ ( 1 - q x ) ⁢ K n Aff ⁡ ( q - x + 1 ; p , N ; q ) = ( 1 - p ) ⁢ ( 1 - q N + 1 ) ⁢ K n + 1 Aff ⁡ ( q - x ; p ⁢ q - 1 , N + 1 ; q ) 1 𝑝 superscript 𝑞 𝑥 1 superscript 𝑞 𝑥 𝑁 1 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 𝑝 1 superscript 𝑞 𝑥 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 1 𝑝 𝑁 𝑞 1 𝑝 1 superscript 𝑞 𝑁 1 affine-q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 superscript 𝑞 1 𝑁 1 𝑞 {\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 (1-pq^x)(1-q^{-x+N+1})\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}-p(1-q^x)\AffqKrawtchouk{n}@{q^{-x+1}}{p}{N}{q} {}=(1-p)(1-q^{N+1})\AffqKrawtchouk{n+1}@{q^{-x}}{pq^{-1}}{N+1}{q} }
∇ ⁡ [ w ⁢ ( x ; p , N ; q ) ⁢ K n Aff ⁡ ( q - x ; p , N ; q ) ] ∇ ⁡ q - x = 1 - q N + 1 1 - q ⁢ w ⁢ ( x ; p ⁢ q - 1 , N + 1 ; q ) ⁢ K n + 1 Aff ⁡ ( q - x ; p ⁢ q - 1 , N + 1 ; q ) ∇ 𝑤 𝑥 𝑝 𝑁 𝑞 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 ∇ superscript 𝑞 𝑥 1 superscript 𝑞 𝑁 1 1 𝑞 𝑤 𝑥 𝑝 superscript 𝑞 1 𝑁 1 𝑞 affine-q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 superscript 𝑞 1 𝑁 1 𝑞 {\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)}}} {\displaystyle \frac{\nabla\left[w(x;p,N;q)\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}\right]}{\nabla q^{-x}} {}=\frac{1-q^{N+1}}{1-q}w(x;pq^{-1},N+1;q)\AffqKrawtchouk{n+1}@{q^{-x}}{pq^{-1}}{N+1}{q} }

Substitution(s): w ⁢ ( x ; p , N ; q ) = ( p ⁢ q ; q ) x ( q ; q ) x ⁢ ( q ; q ) N - x ⁢ p - x 𝑤 𝑥 𝑝 𝑁 𝑞 q-Pochhammer-symbol 𝑝 𝑞 𝑞 𝑥 q-Pochhammer-symbol 𝑞 𝑞 𝑥 q-Pochhammer-symbol 𝑞 𝑞 𝑁 𝑥 superscript 𝑝 𝑥 {\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

w ⁢ ( x ; p , N ; q ) ⁢ K n Aff ⁡ ( q - x ; p , N ; q ) = ( - 1 ) n ⁢ q - N ⁢ n + \binomial ⁢ n ⁢ 2 ⁢ ( 1 - q ) n ( q - N ; q ) n ⁢ ( ∇ q ) n ⁢ [ w ⁢ ( x ; p ⁢ q n , N - n ; q ) ] 𝑤 𝑥 𝑝 𝑁 𝑞 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 superscript 1 𝑛 superscript 𝑞 𝑁 𝑛 \binomial 𝑛 2 superscript 1 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑛 superscript subscript ∇ 𝑞 𝑛 delimited-[] 𝑤 𝑥 𝑝 superscript 𝑞 𝑛 𝑁 𝑛 𝑞 {\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]}}} {\displaystyle w(x;p,N;q)\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q} {}=\frac{(-1)^nq^{-Nn+\binomial{n}{2}}(1-q)^n}{\qPochhammer{q^{-N}}{q}{n}}\left(\nabla_q\right)^n\left[w(x;pq^n,N-n;q)\right] }

Substitution(s): w ⁢ ( x ; p , N ; q ) = ( p ⁢ q ; q ) x ( q ; q ) x ⁢ ( q ; q ) N - x ⁢ p - x 𝑤 𝑥 𝑝 𝑁 𝑞 q-Pochhammer-symbol 𝑝 𝑞 𝑞 𝑥 q-Pochhammer-symbol 𝑞 𝑞 𝑥 q-Pochhammer-symbol 𝑞 𝑞 𝑁 𝑥 superscript 𝑝 𝑥 {\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}}}}


∇ q := ∇ ∇ ⁡ q - x assign subscript ∇ 𝑞 ∇ ∇ superscript 𝑞 𝑥 {\displaystyle{\displaystyle{\displaystyle\nabla_{q}:=\frac{\nabla}{\nabla q^{% -x}}}}} {\displaystyle \nabla_q:=\frac{\nabla}{\nabla q^{-x}} }

Generating functions

( q - N ⁢ t ; q ) N - x ⋅ \qHyperrphis ⁢ 11 ⁢ @ ⁢ @ ⁢ q - x ⁢ p ⁢ q ⁢ q ⁢ p ⁢ q ⁢ t = ∑ n = 0 N ( q - N ; q ) n ( q ; q ) n ⁢ K n Aff ⁡ ( q - x ; p , N ; q ) ⁢ t n ⋅ q-Pochhammer-symbol superscript 𝑞 𝑁 𝑡 𝑞 𝑁 𝑥 \qHyperrphis 11 @ @ superscript 𝑞 𝑥 𝑝 𝑞 𝑞 𝑝 𝑞 𝑡 superscript subscript 𝑛 0 𝑁 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 superscript 𝑡 𝑛 {\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 \qPochhammer{q^{-N}t}{q}{N-x}\cdot\qHyperrphis{1}{1}@@{q^{-x}}{pq}{q}{pqt}=\sum_{n=0}^N \frac{\qPochhammer{q^{-N}}{q}{n}}{\qPochhammer{q}{q}{n}}\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}t^n }
( - p ⁢ q - N + 1 ⁢ t ; q ) x ⋅ \qHyperrphis ⁢ 20 ⁢ @ ⁢ @ ⁢ q x - N , p ⁢ q x + 1 - q - q - x ⁢ t = ∑ n = 0 N ( p ⁢ q , q - N ; q ) n ( q ; q ) n ⁢ q - \binomial ⁢ n ⁢ 2 ⁢ K n Aff ⁡ ( q - x ; p , N ; q ) ⁢ t n ⋅ q-Pochhammer-symbol 𝑝 superscript 𝑞 𝑁 1 𝑡 𝑞 𝑥 \qHyperrphis 20 @ @ superscript 𝑞 𝑥 𝑁 𝑝 superscript 𝑞 𝑥 1 𝑞 superscript 𝑞 𝑥 𝑡 superscript subscript 𝑛 0 𝑁 q-Pochhammer-symbol 𝑝 𝑞 superscript 𝑞 𝑁 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 \binomial 𝑛 2 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 superscript 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}}}} {\displaystyle \qPochhammer{-pq^{-N+1}t}{q}{x}\cdot\qHyperrphis{2}{0}@@{q^{x-N},pq^{x+1}}{-}{q}{-q^{-x}t} {}=\sum_{n=0}^N\frac{\qPochhammer{pq,q^{-N}}{q}{n}}{\qPochhammer{q}{q}{n}}q^{-\binomial{n}{2}}\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}t^n }

Limit relations

q-Hahn polynomial to Affine q-Krawtchouk polynomial

Q n ⁡ ( q - x ; p , 0 , N ; q ) = K n Aff ⁡ ( q - x ; p , N ; q ) q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝑝 0 𝑁 𝑞 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(q^{-x};p,0,N;q\right)=% K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)}}} {\displaystyle \qHahn{n}@{q^{-x}}{p}{0}{N}{q}=\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q} }

Dual q-Hahn polynomial to Affine q-Krawtchouk polynomial

R n ⁡ ( μ ⁢ ( x ) ; p , 0 , N ) ⁢ q = K n Aff ⁡ ( q - x ; p , N ; q ) dual-q-Hahn-R 𝑛 𝜇 𝑥 𝑝 0 𝑁 𝑞 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);p,0,N\right){q}% =K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)}}} {\displaystyle \dualqHahn{n}@{\mu(x)}{p}{0}{N}{q}=\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q} }

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

lim N → ∞ ⁡ K n Aff ⁡ ( q x - N ; p , N ; q ) = p n ⁡ ( q x ; p ; q ) subscript → 𝑁 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑁 𝑝 𝑁 𝑞 little-q-Laguerre-Wall-polynomial-p 𝑛 superscript 𝑞 𝑥 𝑝 𝑞 {\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)}}} {\displaystyle \lim_{N\rightarrow\infty}\AffqKrawtchouk{n}@{q^{x-N}}{p}{N}{q}=\littleqLaguerre{n}@{q^x}{p}{q} }

Affine q-Krawtchouk polynomial to Krawtchouk polynomial

lim q → 1 ⁡ K n Aff ⁡ ( q - x ; p , N ; q ) = K n ⁡ ( x ; 1 - p , N ) subscript → 𝑞 1 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 Krawtchouk-polynomial-K 𝑛 𝑥 1 𝑝 𝑁 {\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)}}} {\displaystyle \lim_{q\rightarrow 1}\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\Krawtchouk{n}@{x}{1-p}{N} }

Remarks

K n Aff ⁡ ( q - x ; p , N ; q ) = P n ⁡ ( q - x ; p , q - N - 1 ; q ) affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 big-q-Laguerre-polynomial-P 𝑛 superscript 𝑞 𝑥 𝑝 superscript 𝑞 𝑁 1 𝑞 {\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 \AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\bigqLaguerre{n}@{q^{-x}}{p}{q^{-N-1}}{q} }
K n Aff ⁡ ( q x ; p , N ; q - 1 ) = 1 ( p - 1 ⁢ q ; q ) n ⁢ K n qtm ⁡ ( q x - N ; p - 1 , N ; q ) affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 superscript 𝑞 1 1 q-Pochhammer-symbol superscript 𝑝 1 𝑞 𝑞 𝑛 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑁 superscript 𝑝 1 𝑁 𝑞 {\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)}}} {\displaystyle \AffqKrawtchouk{n}@{q^x}{p}{N}{q^{-1}}=\frac{1}{\qPochhammer{p^{-1}q}{q}{n}} \qtmqKrawtchouk{n}@{q^{x-N}}{p^{-1}}{N}{q} }

Koornwinder Addendum: Affine q-Krawtchouk

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