Quantum q-Krawtchouk

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Quantum q-Krawtchouk

Basic hypergeometric representation

K n qtm ⁡ ( q - x ; p , N ; q ) = \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - n , q - x ⁢ q - N ⁢ q ⁢ p ⁢ q n + 1 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 \qHyperrphis 21 @ @ superscript 𝑞 𝑛 superscript 𝑞 𝑥 superscript 𝑞 𝑁 𝑞 𝑝 superscript 𝑞 𝑛 1 {\displaystyle{\displaystyle{\displaystyle K^{\mathrm{qtm}}_{n}\!\left(q^{-x};% p,N;q\right)=\qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{q^{-N}}{q}{pq^{n+1}}}}} {\displaystyle \qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}= \qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{q^{-N}}{q}{pq^{n+1}} }

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


Orthogonality relation(s)

∑ x = 0 N ( p ⁢ q ; q ) N - x ( q ; q ) x ⁢ ( q ; q ) N - x ⁢ ( - 1 ) N - x ⁢ q \binomial ⁢ x ⁢ 2 ⁢ K m qtm ⁡ ( q - x ; p , N ; q ) ⁢ K n qtm ⁡ ( q - x ; p , N ; q ) = ( - 1 ) n ⁢ p N ⁢ ( q ; q ) N - n ⁢ ( q , p ⁢ q ; q ) n ( q , q ; q ) N ⁢ q \binomial ⁢ N + 12 - \binomial ⁢ n + 12 + N ⁢ n ⁢ δ m , n superscript subscript 𝑥 0 𝑁 q-Pochhammer-symbol 𝑝 𝑞 𝑞 𝑁 𝑥 q-Pochhammer-symbol 𝑞 𝑞 𝑥 q-Pochhammer-symbol 𝑞 𝑞 𝑁 𝑥 superscript 1 𝑁 𝑥 superscript 𝑞 \binomial 𝑥 2 quantum-q-Krawtchouk-polynomial-K 𝑚 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 superscript 1 𝑛 superscript 𝑝 𝑁 q-Pochhammer-symbol 𝑞 𝑞 𝑁 𝑛 q-Pochhammer-symbol 𝑞 𝑝 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑞 𝑁 superscript 𝑞 \binomial 𝑁 12 \binomial 𝑛 12 𝑁 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\frac{\left(pq;q\right% )_{N-x}}{\left(q;q\right)_{x}\left(q;q\right)_{N-x}}(-1)^{N-x}q^{\binomial{x}{% 2}}K^{\mathrm{qtm}}_{m}\!\left(q^{-x};p,N;q\right)K^{\mathrm{qtm}}_{n}\!\left(% q^{-x};p,N;q\right){}=\frac{(-1)^{n}p^{N}\left(q;q\right)_{N-n}\left(q,pq;q% \right)_{n}}{\left(q,q;q\right)_{N}}q^{\binomial{N+1}{2}-\binomial{n+1}{2}+Nn}% \,\delta_{m,n}}}} {\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): p > q - N 𝑝 superscript 𝑞 𝑁 {\displaystyle{\displaystyle{\displaystyle p>q^{-N}}}}


Recurrence relation

- p ⁢ q 2 ⁢ n + 1 ⁢ ( 1 - q - x ) ⁢ K n qtm ⁡ ( q - x ) = ( 1 - q n - N ) ⁢ K n + 1 qtm ⁡ ( q - x ) - [ ( 1 - q n - N ) + q ⁢ ( 1 - q n ) ⁢ ( 1 - p ⁢ q n ) ] ⁢ K n qtm ⁡ ( q - x ) + q ⁢ ( 1 - q n ) ⁢ ( 1 - p ⁢ q n ) ⁢ K n - 1 qtm ⁡ ( q - x ) 𝑝 superscript 𝑞 2 𝑛 1 1 superscript 𝑞 𝑥 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 1 superscript 𝑞 𝑛 𝑁 quantum-q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 delimited-[] 1 superscript 𝑞 𝑛 𝑁 𝑞 1 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 𝑛 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 𝑞 1 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 𝑛 quantum-q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle-pq^{2n+1}(1-q^{-x})K^{\mathrm{qtm}}% _{n}\!\left(q^{-x}\right){}=(1-q^{n-N})K^{\mathrm{qtm}}_{n+1}\!\left(q^{-x}% \right){}-\left[(1-q^{n-N})+q(1-q^{n})(1-pq^{n})\right]K^{\mathrm{qtm}}_{n}\!% \left(q^{-x}\right){}+q(1-q^{n})(1-pq^{n})K^{\mathrm{qtm}}_{n-1}\!\left(q^{-x}% \right)}}} {\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} }
K n qtm ⁡ ( q - x ) := K n qtm ⁡ ( q - x ; p , N ; q ) assign quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle K^{\mathrm{qtm}}_{n}\!\left(q^{-x}% \right):=K^{\mathrm{qtm}}_{n}\!\left(q^{-x};p,N;q\right)}}} {\displaystyle \qtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q}:=\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q} }

Monic recurrence relation

x ⁢ K ^ qtm n ⁡ ( x ) = K ^ qtm n + 1 ⁡ ( x ) + [ 1 - p - 1 ⁢ q - 2 ⁢ n - 1 ⁢ { ( 1 - q n - N ) + q ⁢ ( 1 - q n ) ⁢ ( 1 - p ⁢ q n ) } ] ⁢ K ^ qtm n ⁡ ( x ) + p - 2 ⁢ q - 4 ⁢ n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 - p ⁢ q n ) ⁢ ( 1 - q n - N - 1 ) ⁢ K ^ qtm n - 1 ⁡ ( x ) 𝑥 quantum-q-Krawtchouk-polynomial-monic-p 𝑛 𝑥 𝑝 𝑁 𝑞 quantum-q-Krawtchouk-polynomial-monic-p 𝑛 1 𝑥 𝑝 𝑁 𝑞 delimited-[] 1 superscript 𝑝 1 superscript 𝑞 2 𝑛 1 1 superscript 𝑞 𝑛 𝑁 𝑞 1 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 𝑛 quantum-q-Krawtchouk-polynomial-monic-p 𝑛 𝑥 𝑝 𝑁 𝑞 superscript 𝑝 2 superscript 𝑞 4 𝑛 1 1 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 𝑁 1 quantum-q-Krawtchouk-polynomial-monic-p 𝑛 1 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{K}^{\mathrm{qtm}}}_{n}\!% \left(x\right)={\widehat{K}^{\mathrm{qtm}}}_{n+1}\!\left(x\right)+\left[1-p^{-% 1}q^{-2n-1}\left\{(1-q^{n-N})+q(1-q^{n})(1-pq^{n})\right\}\right]{\widehat{K}^% {\mathrm{qtm}}}_{n}\!\left(x\right){}+p^{-2}q^{-4n+1}(1-q^{n})(1-pq^{n})(1-q^{% n-N-1}){\widehat{K}^{\mathrm{qtm}}}_{n-1}\!\left(x\right)}}} {\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} }
K n qtm ⁡ ( q - x ; p , N ; q ) = p n ⁢ q n 2 ( q - N ; q ) n ⁢ K ^ qtm n ⁡ ( q - x ) quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 superscript 𝑝 𝑛 superscript 𝑞 superscript 𝑛 2 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑛 quantum-q-Krawtchouk-polynomial-monic-p 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle K^{\mathrm{qtm}}_{n}\!\left(q^{-x};% p,N;q\right)=\frac{p^{n}q^{n^{2}}}{\left(q^{-N};q\right)_{n}}{\widehat{K}^{% \mathrm{qtm}}}_{n}\!\left(q^{-x}\right)}}} {\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

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

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

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


Forward shift operator

K n qtm ⁡ ( q - x - 1 ; p , N ; q ) - K n qtm ⁡ ( q - x ; p , N ; q ) = p ⁢ q - x ⁢ ( 1 - q n ) 1 - q - N ⁢ K n - 1 qtm ⁡ ( q - x ; p ⁢ q , N - 1 ; q ) quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 1 𝑝 𝑁 𝑞 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 𝑝 superscript 𝑞 𝑥 1 superscript 𝑞 𝑛 1 superscript 𝑞 𝑁 quantum-q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 𝑞 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle K^{\mathrm{qtm}}_{n}\!\left(q^{-x-1% };p,N;q\right)-K^{\mathrm{qtm}}_{n}\!\left(q^{-x};p,N;q\right){}=\frac{pq^{-x}% (1-q^{n})}{1-q^{-N}}K^{\mathrm{qtm}}_{n-1}\!\left(q^{-x};pq,N-1;q\right)}}} {\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} }
Δ ⁢ K n qtm ⁡ ( q - x ; p , N ; q ) Δ ⁢ q - x = p ⁢ q ⁢ ( 1 - q n ) ( 1 - q ) ⁢ ( 1 - q - N ) ⁢ K n - 1 qtm ⁡ ( q - x ; p ⁢ q , N - 1 ; q ) Δ quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 Δ superscript 𝑞 𝑥 𝑝 𝑞 1 superscript 𝑞 𝑛 1 𝑞 1 superscript 𝑞 𝑁 quantum-q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 𝑞 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{\Delta K^{\mathrm{qtm}}_{n}\!% \left(q^{-x};p,N;q\right)}{\Delta q^{-x}}=\frac{pq(1-q^{n})}{(1-q)(1-q^{-N})}K% ^{\mathrm{qtm}}_{n-1}\!\left(q^{-x};pq,N-1;q\right)}}} {\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

( 1 - q x - N - 1 ) ⁢ K n qtm ⁡ ( q - x ; p , N ; q ) + q - x ⁢ ( 1 - q x ) ⁢ ( p - q x - N - 1 ) ⁢ K n qtm ⁡ ( q - x + 1 ; p , N ; q ) = ( 1 - q - N - 1 ) ⁢ K n + 1 qtm ⁡ ( q - x ; p ⁢ q - 1 , N + 1 ; q ) 1 superscript 𝑞 𝑥 𝑁 1 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 superscript 𝑞 𝑥 1 superscript 𝑞 𝑥 𝑝 superscript 𝑞 𝑥 𝑁 1 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 1 𝑝 𝑁 𝑞 1 superscript 𝑞 𝑁 1 quantum-q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 superscript 𝑞 1 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle(1-q^{x-N-1})K^{\mathrm{qtm}}_{n}\!% \left(q^{-x};p,N;q\right){}+q^{-x}(1-q^{x})(p-q^{x-N-1})K^{\mathrm{qtm}}_{n}\!% \left(q^{-x+1};p,N;q\right){}=(1-q^{-N-1})K^{\mathrm{qtm}}_{n+1}\!\left(q^{-x}% ;pq^{-1},N+1;q\right)}}} {\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} }
∇ ⁡ [ w ⁢ ( x ; p , N ; q ) ⁢ K n qtm ⁡ ( q - x ; p , N ; q ) ] ∇ ⁡ q - x = 1 1 - q ⁢ w ⁢ ( x ; p ⁢ q - 1 , N + 1 ; q ) ⁢ K n + 1 qtm ⁡ ( q - x ; p ⁢ q - 1 , N + 1 ; q ) ∇ 𝑤 𝑥 𝑝 𝑁 𝑞 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 ∇ superscript 𝑞 𝑥 1 1 𝑞 𝑤 𝑥 𝑝 superscript 𝑞 1 𝑁 1 𝑞 quantum-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{qtm}}_{n}\!\left(q^{-x};p,N;q\right)\right]}{\nabla q^{-x}}{}=\frac{1}% {1-q}w(x;pq^{-1},N+1;q)K^{\mathrm{qtm}}_{n+1}\!\left(q^{-x};pq^{-1},N+1;q% \right)}}} {\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): w ⁢ ( x ; p , N ; q ) = ( q - N ; q ) x ( q , p - 1 ⁢ q - N ; q ) x ⁢ ( - p ) - x ⁢ q \binomial ⁢ x + 12 𝑤 𝑥 𝑝 𝑁 𝑞 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑥 q-Pochhammer-symbol 𝑞 superscript 𝑝 1 superscript 𝑞 𝑁 𝑞 𝑥 superscript 𝑝 𝑥 superscript 𝑞 \binomial 𝑥 12 {\displaystyle{\displaystyle{\displaystyle w(x;p,N;q)=\frac{\left(q^{-N};q% \right)_{x}}{\left(q,p^{-1}q^{-N};q\right)_{x}}(-p)^{-x}q^{\binomial{x+1}{2}}}}}


Rodrigues-type formula

w ⁢ ( x ; p , N ; q ) ⁢ K n qtm ⁡ ( q - x ; p , N ; q ) = ( 1 - q ) n ⁢ ( ∇ q ) n ⁢ [ w ⁢ ( x ; p ⁢ q n , N - n ; q ) ] 𝑤 𝑥 𝑝 𝑁 𝑞 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 superscript 1 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{qtm}}_{n}\!% \left(q^{-x};p,N;q\right)=(1-q)^{n}\left(\nabla_{q}\right)^{n}\left[w(x;pq^{n}% ,N-n;q)\right]}}} {\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): w ⁢ ( x ; p , N ; q ) = ( q - N ; q ) x ( q , p - 1 ⁢ q - N ; q ) x ⁢ ( - p ) - x ⁢ q \binomial ⁢ x + 12 𝑤 𝑥 𝑝 𝑁 𝑞 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑥 q-Pochhammer-symbol 𝑞 superscript 𝑝 1 superscript 𝑞 𝑁 𝑞 𝑥 superscript 𝑝 𝑥 superscript 𝑞 \binomial 𝑥 12 {\displaystyle{\displaystyle{\displaystyle w(x;p,N;q)=\frac{\left(q^{-N};q% \right)_{x}}{\left(q,p^{-1}q^{-N};q\right)_{x}}(-p)^{-x}q^{\binomial{x+1}{2}}}}}


∇ 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 x - N ⁢ t ; q ) N - x ⋅ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - x , p ⁢ q N + 1 - x ⁢ 0 ⁢ q ⁢ q x - N ⁢ t = ∑ n = 0 N ( q - N ; q ) n ( q ; q ) n ⁢ K n qtm ⁡ ( q - x ; p , N ; q ) ⁢ t n ⋅ q-Pochhammer-symbol superscript 𝑞 𝑥 𝑁 𝑡 𝑞 𝑁 𝑥 \qHyperrphis 21 @ @ superscript 𝑞 𝑥 𝑝 superscript 𝑞 𝑁 1 𝑥 0 𝑞 superscript 𝑞 𝑥 𝑁 𝑡 superscript subscript 𝑛 0 𝑁 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left(q^{x-N}t;q\right)_{N-x}\cdot% \qHyperrphis{2}{1}@@{q^{-x},pq^{N+1-x}}{0}{q}{q^{x-N}t}{}=\sum_{n=0}^{N}\frac{% \left(q^{-N};q\right)_{n}}{\left(q;q\right)_{n}}K^{\mathrm{qtm}}_{n}\!\left(q^% {-x};p,N;q\right)t^{n}}}} {\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 }
( q - x ⁢ t ; q ) x ⋅ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q x - N , 0 ⁢ p ⁢ q ⁢ q ⁢ q - x ⁢ t = ∑ n = 0 N ( q - N ; q ) n ( p ⁢ q , q ; q ) n ⁢ K n qtm ⁡ ( q - x ; p , N ; q ) ⁢ t n ⋅ q-Pochhammer-symbol superscript 𝑞 𝑥 𝑡 𝑞 𝑥 \qHyperrphis 21 @ @ superscript 𝑞 𝑥 𝑁 0 𝑝 𝑞 𝑞 superscript 𝑞 𝑥 𝑡 superscript subscript 𝑛 0 𝑁 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑛 q-Pochhammer-symbol 𝑝 𝑞 𝑞 𝑞 𝑛 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left(q^{-x}t;q\right)_{x}\cdot% \qHyperrphis{2}{1}@@{q^{x-N},0}{pq}{q}{q^{-x}t}{}=\sum_{n=0}^{N}\frac{\left(q^% {-N};q\right)_{n}}{\left(pq,q;q\right)_{n}}K^{\mathrm{qtm}}_{n}\!\left(q^{-x};% p,N;q\right)t^{n}}}} {\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

lim α → ∞ ⁡ Q n ⁡ ( q - x ; α , p , N ; q ) = K n qtm ⁡ ( q - x ; p , N ; q ) subscript → 𝛼 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝑝 𝑁 𝑞 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}Q_{n}% \!\left(q^{-x};\alpha,p,N;q\right)=K^{\mathrm{qtm}}_{n}\!\left(q^{-x};p,N;q% \right)}}} {\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

lim N → ∞ ⁡ K n qtm ⁡ ( x ; a - 1 ⁢ q - N - 1 , N ; q ) = ( - 1 a ) n ⁢ q \binomial ⁢ n ⁢ 2 ⁢ V n ( a ) ⁡ ( x ; q ) subscript → 𝑁 quantum-q-Krawtchouk-polynomial-K 𝑛 𝑥 superscript 𝑎 1 superscript 𝑞 𝑁 1 𝑁 𝑞 superscript 1 𝑎 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Al-Salam-Carlitz-II-polynomial-V 𝑎 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}K^{\mathrm{% qtm}}_{n}\!\left(x;a^{-1}q^{-N-1},N;q\right)=\left(-\frac{1}{a}\right)^{n}q^{% \binomial{n}{2}}V^{(a)}_{n}\!\left(x;q\right)}}} {\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

lim q → 1 ⁡ K n qtm ⁡ ( q - x ; p , N ; q ) = K n ⁡ ( x ; p - 1 , N ) subscript → 𝑞 1 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 Krawtchouk-polynomial-K 𝑛 𝑥 superscript 𝑝 1 𝑁 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}K^{\mathrm{qtm}% }_{n}\!\left(q^{-x};p,N;q\right)=K_{n}\!\left(x;p^{-1},N\right)}}} {\displaystyle \lim_{q\rightarrow 1}\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\Krawtchouk{n}@{x}{p^{-1}}{N} }

Remarks

K n qtm ⁡ ( q - x ; p , N ; q ) = M n ⁡ ( q - x ; q - N - 1 , - p - 1 ; q ) quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 q-Meixner-polynomial-M 𝑛 superscript 𝑞 𝑥 superscript 𝑞 𝑁 1 superscript 𝑝 1 𝑞 {\displaystyle{\displaystyle{\displaystyle K^{\mathrm{qtm}}_{n}\!\left(q^{-x};% p,N;q\right)=M_{n}\!\left(q^{-x};q^{-N-1},-p^{-1};q\right)}}} {\displaystyle \qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\qMeixner{n}@{q^{-x}}{q^{-N-1}}{-p^{-1}}{q} }
K n qtm ⁡ ( q x ; p , N ; q - 1 ) = ( p - 1 ⁢ q ; q ) n ⁢ ( - p q ) n ⁢ q - \binomial ⁢ n ⁢ 2 ⁢ K n Aff ⁡ ( q x - N ; p - 1 , N ; q ) quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 superscript 𝑞 1 q-Pochhammer-symbol superscript 𝑝 1 𝑞 𝑞 𝑛 superscript 𝑝 𝑞 𝑛 superscript 𝑞 \binomial 𝑛 2 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑁 superscript 𝑝 1 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle K^{\mathrm{qtm}}_{n}\!\left(q^{x};p% ,N;q^{-1}\right)=\left(p^{-1}q;q\right)_{n}\left(-\frac{p}{q}\right)^{n}q^{-% \binomial{n}{2}}K^{\mathrm{Aff}}_{n}\!\left(q^{x-N};p^{-1},N;q\right)}}} {\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} }

Koornwinder Addendum: Quantum q-Krawtchouk

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