q-Krawtchouk

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q-Krawtchouk

Basic hypergeometric representation

K n ⁡ ( q - x ; p , N ; q ) = \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ q - n , q - x , - p ⁢ q n ⁢ q - N , 0 ⁢ q ⁢ q q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 superscript 𝑞 𝑥 𝑝 superscript 𝑞 𝑛 superscript 𝑞 𝑁 0 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(q^{-x};p,N;q\right)=% \qHyperrphis{3}{2}@@{q^{-n},q^{-x},-pq^{n}}{q^{-N},0}{q}{q}}}} {\displaystyle \qKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\qHyperrphis{3}{2}@@{q^{-n},q^{-x},-pq^n}{q^{-N},0}{q}{q} }
K n ⁡ ( q - x ; p , N ; q ) = ( q x - N ; q ) n ( q - N ; q ) n ⁢ q n ⁢ x ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - n , q - x ⁢ q N - x - n + 1 ⁢ q - p ⁢ q n + N + 1 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 q-Pochhammer-symbol superscript 𝑞 𝑥 𝑁 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑛 superscript 𝑞 𝑛 𝑥 \qHyperrphis 21 @ @ superscript 𝑞 𝑛 superscript 𝑞 𝑥 superscript 𝑞 𝑁 𝑥 𝑛 1 𝑞 𝑝 superscript 𝑞 𝑛 𝑁 1 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(q^{-x};p,N;q\right)=% \frac{\left(q^{x-N};q\right)_{n}}{\left(q^{-N};q\right)_{n}q^{nx}}\ % \qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{q^{N-x-n+1}}{q}{-pq^{n+N+1}}}}} {\displaystyle \qKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\frac{\qPochhammer{q^{x-N}}{q}{n}}{\qPochhammer{q^{-N}}{q}{n}q^{nx}}\ \qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{q^{N-x-n+1}}{q}{-pq^{n+N+1}} }

Orthogonality relation(s)

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

Constraint(s): p > 0 𝑝 0 {\displaystyle{\displaystyle{\displaystyle p>0}}}


Recurrence relation

- ( 1 - q - x ) ⁢ K n ⁡ ( q - x ) = A n ⁢ K n + 1 ⁡ ( q - x ) - ( A n + C n ) ⁢ K n ⁡ ( q - x ) + C n ⁢ K n - 1 ⁡ ( q - x ) 1 superscript 𝑞 𝑥 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 subscript 𝐴 𝑛 q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 subscript 𝐶 𝑛 q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle-\left(1-q^{-x}\right)K_{n}\!\left(q% ^{-x}\right)=A_{n}K_{n+1}\!\left(q^{-x}\right)-\left(A_{n}+C_{n}\right)K_{n}\!% \left(q^{-x}\right){}+C_{n}K_{n-1}\!\left(q^{-x}\right)}}} {\displaystyle -\left(1-q^{-x}\right)\qKrawtchouk{n}@@{q^{-x}}{p}{N}{q}=A_n\qKrawtchouk{n+1}@@{q^{-x}}{p}{N}{q}-\left(A_n+C_n\right)\qKrawtchouk{n}@@{q^{-x}}{p}{N}{q} {}+C_n\qKrawtchouk{n-1}@@{q^{-x}}{p}{N}{q} }

Substitution(s): C n = - p ⁢ q 2 ⁢ n - N - 1 ⁢ ( 1 + p ⁢ q n + N ) ⁢ ( 1 - q n ) ( 1 + p ⁢ q 2 ⁢ n - 1 ) ⁢ ( 1 + p ⁢ q 2 ⁢ n ) subscript 𝐶 𝑛 𝑝 superscript 𝑞 2 𝑛 𝑁 1 1 𝑝 superscript 𝑞 𝑛 𝑁 1 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 2 𝑛 1 1 𝑝 superscript 𝑞 2 𝑛 {\displaystyle{\displaystyle{\displaystyle C_{n}=-pq^{2n-N-1}\frac{(1+pq^{n+N}% )(1-q^{n})}{(1+pq^{2n-1})(1+pq^{2n})}}}} &
A n = ( 1 - q n - N ) ⁢ ( 1 + p ⁢ q n ) ( 1 + p ⁢ q 2 ⁢ n ) ⁢ ( 1 + p ⁢ q 2 ⁢ n + 1 ) subscript 𝐴 𝑛 1 superscript 𝑞 𝑛 𝑁 1 𝑝 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 2 𝑛 1 𝑝 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n-N})(1+pq^{n})}{% (1+pq^{2n})(1+pq^{2n+1})}}}}


K n ⁡ ( q - x ) := K n ⁡ ( q - x ; p , N ; q ) assign q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(q^{-x}\right):=K_{n}\!% \left(q^{-x};p,N;q\right)}}} {\displaystyle \qKrawtchouk{n}@@{q^{-x}}{p}{N}{q}:=\qKrawtchouk{n}@{q^{-x}}{p}{N}{q} }

Monic recurrence relation

x ⁢ K ^ n ⁡ ( x ) = K ^ n + 1 ⁡ ( x ) + [ 1 - ( A n + C n ) ] ⁢ K ^ n ⁡ ( x ) + A n - 1 ⁢ C n ⁢ K ^ n - 1 ⁡ ( x ) 𝑥 q-Krawtchouk-polynomial-monic-p 𝑛 𝑥 𝑝 𝑁 𝑞 q-Krawtchouk-polynomial-monic-p 𝑛 1 𝑥 𝑝 𝑁 𝑞 delimited-[] 1 subscript 𝐴 𝑛 subscript 𝐶 𝑛 q-Krawtchouk-polynomial-monic-p 𝑛 𝑥 𝑝 𝑁 𝑞 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 q-Krawtchouk-polynomial-monic-p 𝑛 1 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{K}}_{n}\!\left(x\right)=% {\widehat{K}}_{n+1}\!\left(x\right)+\left[1-(A_{n}+C_{n})\right]{\widehat{K}}_% {n}\!\left(x\right)+A_{n-1}C_{n}{\widehat{K}}_{n-1}\!\left(x\right)}}} {\displaystyle x\monicqKrawtchouk{n}@@{x}{p}{N}{q}=\monicqKrawtchouk{n+1}@@{x}{p}{N}{q}+\left[1-(A_n+C_n)\right]\monicqKrawtchouk{n}@@{x}{p}{N}{q}+A_{n-1}C_n\monicqKrawtchouk{n-1}@@{x}{p}{N}{q} }

Substitution(s): C n = - p ⁢ q 2 ⁢ n - N - 1 ⁢ ( 1 + p ⁢ q n + N ) ⁢ ( 1 - q n ) ( 1 + p ⁢ q 2 ⁢ n - 1 ) ⁢ ( 1 + p ⁢ q 2 ⁢ n ) subscript 𝐶 𝑛 𝑝 superscript 𝑞 2 𝑛 𝑁 1 1 𝑝 superscript 𝑞 𝑛 𝑁 1 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 2 𝑛 1 1 𝑝 superscript 𝑞 2 𝑛 {\displaystyle{\displaystyle{\displaystyle C_{n}=-pq^{2n-N-1}\frac{(1+pq^{n+N}% )(1-q^{n})}{(1+pq^{2n-1})(1+pq^{2n})}}}} &
A n = ( 1 - q n - N ) ⁢ ( 1 + p ⁢ q n ) ( 1 + p ⁢ q 2 ⁢ n ) ⁢ ( 1 + p ⁢ q 2 ⁢ n + 1 ) subscript 𝐴 𝑛 1 superscript 𝑞 𝑛 𝑁 1 𝑝 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 2 𝑛 1 𝑝 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n-N})(1+pq^{n})}{% (1+pq^{2n})(1+pq^{2n+1})}}}}


K n ⁡ ( q - x ; p , N ; q ) = ( - p ⁢ q n ; q ) n ( q - N ; q ) n ⁢ K ^ n ⁡ ( q - x ) q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 q-Pochhammer-symbol 𝑝 superscript 𝑞 𝑛 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑛 q-Krawtchouk-polynomial-monic-p 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(q^{-x};p,N;q\right)=% \frac{\left(-pq^{n};q\right)_{n}}{\left(q^{-N};q\right)_{n}}{\widehat{K}}_{n}% \!\left(q^{-x}\right)}}} {\displaystyle \qKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\frac{\qPochhammer{-pq^n}{q}{n}}{\qPochhammer{q^{-N}}{q}{n}}\monicqKrawtchouk{n}@@{q^{-x}}{p}{N}{q} }

q-Difference equation

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

Substitution(s): y ⁢ ( x ) = K n ⁡ ( q - x ; p , N ; q ) 𝑦 𝑥 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=K_{n}\!\left(q^{-x};p,N;q% \right)}}}


Forward shift operator

K n ⁡ ( q - x - 1 ; p , N ; q ) - K n ⁡ ( q - x ; p , N ; q ) = q - n - x ⁢ ( 1 - q n ) ⁢ ( 1 + p ⁢ q n ) 1 - q - N ⁢ K n - 1 ⁡ ( q - x ; p ⁢ q 2 , N - 1 ; q ) q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 1 𝑝 𝑁 𝑞 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 superscript 𝑞 𝑛 𝑥 1 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 𝑛 1 superscript 𝑞 𝑁 q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 superscript 𝑞 2 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(q^{-x-1};p,N;q\right)-% K_{n}\!\left(q^{-x};p,N;q\right){}=\frac{q^{-n-x}(1-q^{n})(1+pq^{n})}{1-q^{-N}% }K_{n-1}\!\left(q^{-x};pq^{2},N-1;q\right)}}} {\displaystyle \qKrawtchouk{n}@{q^{-x-1}}{p}{N}{q}-\qKrawtchouk{n}@{q^{-x}}{p}{N}{q} {}=\frac{q^{-n-x}(1-q^n)(1+pq^n)}{1-q^{-N}}\qKrawtchouk{n-1}@{q^{-x}}{pq^2}{N-1}{q} }
Δ ⁢ K n ⁡ ( q - x ; p , N ; q ) Δ ⁢ q - x = q - n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 + p ⁢ q n ) ( 1 - q ) ⁢ ( 1 - q - N ) ⁢ K n - 1 ⁡ ( q - x ; p ⁢ q 2 , N - 1 ; q ) Δ q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 Δ superscript 𝑞 𝑥 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 𝑛 1 𝑞 1 superscript 𝑞 𝑁 q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 superscript 𝑞 2 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{\Delta K_{n}\!\left(q^{-x};p,N% ;q\right)}{\Delta q^{-x}}=\frac{q^{-n+1}(1-q^{n})(1+pq^{n})}{(1-q)(1-q^{-N})}K% _{n-1}\!\left(q^{-x};pq^{2},N-1;q\right)}}} {\displaystyle \frac{\Delta \qKrawtchouk{n}@{q^{-x}}{p}{N}{q}}{\Delta q^{-x}}= \frac{q^{-n+1}(1-q^n)(1+pq^n)}{(1-q)(1-q^{-N})}\qKrawtchouk{n-1}@{q^{-x}}{pq^2}{N-1}{q} }

Backward shift operator

( 1 - q x - N - 1 ) ⁢ K n ⁡ ( q - x ; p , N ; q ) + p ⁢ q - 1 ⁢ ( 1 - q x ) ⁢ K n ⁡ ( q - x + 1 ; p , N ; q ) = q x ⁢ ( 1 - q - N - 1 ) ⁢ K n + 1 ⁡ ( q - x ; p ⁢ q - 2 , N + 1 ; q ) 1 superscript 𝑞 𝑥 𝑁 1 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 𝑝 superscript 𝑞 1 1 superscript 𝑞 𝑥 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 1 𝑝 𝑁 𝑞 superscript 𝑞 𝑥 1 superscript 𝑞 𝑁 1 q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 superscript 𝑞 2 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle(1-q^{x-N-1})K_{n}\!\left(q^{-x};p,N% ;q\right)+pq^{-1}(1-q^{x})K_{n}\!\left(q^{-x+1};p,N;q\right){}=q^{x}(1-q^{-N-1% })K_{n+1}\!\left(q^{-x};pq^{-2},N+1;q\right)}}} {\displaystyle (1-q^{x-N-1})\qKrawtchouk{n}@{q^{-x}}{p}{N}{q}+pq^{-1}(1-q^x)\qKrawtchouk{n}@{q^{-x+1}}{p}{N}{q} {}=q^x(1-q^{-N-1})\qKrawtchouk{n+1}@{q^{-x}}{pq^{-2}}{N+1}{q} }
∇ ⁡ [ w ⁢ ( x ; p , N ; q ) ⁢ K n ⁡ ( q - x ; p , N ; q ) ] ∇ ⁡ q - x = 1 1 - q ⁢ w ⁢ ( x ; p ⁢ q - 2 , N + 1 ; q ) ⁢ K n + 1 ⁡ ( q - x ; p ⁢ q - 2 , N + 1 ; q ) ∇ 𝑤 𝑥 𝑝 𝑁 𝑞 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 ∇ superscript 𝑞 𝑥 1 1 𝑞 𝑤 𝑥 𝑝 superscript 𝑞 2 𝑁 1 𝑞 q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 superscript 𝑞 2 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{\nabla\left[w(x;p,N;q)K_{n}\!% \left(q^{-x};p,N;q\right)\right]}{\nabla q^{-x}}{}=\frac{1}{1-q}w(x;pq^{-2},N+% 1;q)K_{n+1}\!\left(q^{-x};pq^{-2},N+1;q\right)}}} {\displaystyle \frac{\nabla\left[w(x;p,N;q)\qKrawtchouk{n}@{q^{-x}}{p}{N}{q}\right]}{\nabla q^{-x}} {}=\frac{1}{1-q}w(x;pq^{-2},N+1;q)\qKrawtchouk{n+1}@{q^{-x}}{pq^{-2}}{N+1}{q} }

Substitution(s): w ⁢ ( x ; p , N ; q ) = ( q - N ; q ) x ( q ; q ) x ⁢ ( - q p ) x 𝑤 𝑥 𝑝 𝑁 𝑞 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑥 q-Pochhammer-symbol 𝑞 𝑞 𝑥 superscript 𝑞 𝑝 𝑥 {\displaystyle{\displaystyle{\displaystyle w(x;p,N;q)=\frac{\left(q^{-N};q% \right)_{x}}{\left(q;q\right)_{x}}\left(-\frac{q}{p}\right)^{x}}}}


Rodrigues-type formula

w ⁢ ( x ; p , N ; q ) ⁢ K n ⁡ ( q - x ; p , N ; q ) = ( 1 - q ) n ⁢ ( ∇ q ) n ⁢ [ w ⁢ ( x ; p ⁢ q 2 ⁢ n , N - n ; q ) ] 𝑤 𝑥 𝑝 𝑁 𝑞 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 2 𝑛 𝑁 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle w(x;p,N;q)K_{n}\!\left(q^{-x};p,N;q% \right)=(1-q)^{n}\left(\nabla_{q}\right)^{n}\left[w(x;pq^{2n},N-n;q)\right]}}} {\displaystyle w(x;p,N;q)\qKrawtchouk{n}@{q^{-x}}{p}{N}{q}=(1-q)^n\left(\nabla_q\right)^n\left[w(x;pq^{2n},N-n;q)\right] }

Substitution(s): w ⁢ ( x ; p , N ; q ) = ( q - N ; q ) x ( q ; q ) x ⁢ ( - q p ) x 𝑤 𝑥 𝑝 𝑁 𝑞 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑥 q-Pochhammer-symbol 𝑞 𝑞 𝑥 superscript 𝑞 𝑝 𝑥 {\displaystyle{\displaystyle{\displaystyle w(x;p,N;q)=\frac{\left(q^{-N};q% \right)_{x}}{\left(q;q\right)_{x}}\left(-\frac{q}{p}\right)^{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 function

\qHyperrphis ⁢ 11 ⁢ @ ⁢ @ ⁢ q - x ⁢ 0 ⁢ q ⁢ p ⁢ q ⁢ t ⁢ \qHyperrphis ⁢ 20 ⁢ @ ⁢ @ ⁢ q x - N , 0 - q - q - x ⁢ t = ∑ n = 0 N ( q - N ; q ) n ( q ; q ) n ⁢ q - \binomial ⁢ n ⁢ 2 ⁢ K n ⁡ ( q - x ; p , N ; q ) ⁢ t n \qHyperrphis 11 @ @ superscript 𝑞 𝑥 0 𝑞 𝑝 𝑞 𝑡 \qHyperrphis 20 @ @ superscript 𝑞 𝑥 𝑁 0 𝑞 superscript 𝑞 𝑥 𝑡 superscript subscript 𝑛 0 𝑁 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\qHyperrphis{1}{1}@@{q^{-x}}{0}{q}{% pqt}\,\qHyperrphis{2}{0}@@{q^{x-N},0}{-}{q}{-q^{-x}t}{}=\sum_{n=0}^{N}\frac{% \left(q^{-N};q\right)_{n}}{\left(q;q\right)_{n}}q^{-\binomial{n}{2}}K_{n}\!% \left(q^{-x};p,N;q\right)t^{n}}}} {\displaystyle \qHyperrphis{1}{1}@@{q^{-x}}{0}{q}{pqt}\,\qHyperrphis{2}{0}@@{q^{x-N},0}{-}{q}{-q^{-x}t} {}=\sum_{n=0}^N\frac{\qPochhammer{q^{-N}}{q}{n}}{\qPochhammer{q}{q}{n}}q^{-\binomial{n}{2}}\qKrawtchouk{n}@{q^{-x}}{p}{N}{q}t^n }

Limit relations

q-Racah polynomial to q-Krawtchouk polynomial

R n ⁡ ( q - x ; q - N - 1 , - p ⁢ q N , 0 , 0 | q ) = K n ⁡ ( q - x ; p , N ; q ) q-Racah-polynomial-R 𝑛 superscript 𝑞 𝑥 superscript 𝑞 𝑁 1 𝑝 superscript 𝑞 𝑁 0 0 𝑞 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(q^{-x};q^{-N-1},-pq^{N% },0,0\,|\,q\right)=K_{n}\!\left(q^{-x};p,N;q\right)}}} {\displaystyle \qRacah{n}@{q^{-x}}{q^{-N-1}}{-pq^N}{0}{0}{q}=\qKrawtchouk{n}@{q^{-x}}{p}{N}{q} }

q-Hahn polynomial to q-Krawtchouk polynomial

lim α → 0 ⁡ Q n ⁢ ( q - x ; α , - α - 1 ⁢ q - 1 ⁢ p , N | q ) = K n ⁡ ( q - x ; p , N ; q ) subscript → 𝛼 0 subscript 𝑄 𝑛 superscript 𝑞 𝑥 𝛼 superscript 𝛼 1 superscript 𝑞 1 𝑝 conditional 𝑁 𝑞 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow 0}Q_{n}(q^{-% x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=K_{n}\!\left(q^{-x};p,N;q\right)}}} {\displaystyle \lim_{\alpha\rightarrow 0} Q_n(q^{-x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=\qKrawtchouk{n}@{q^{-x}}{p}{N}{q} }

q-Krawtchouk polynomial to q-Bessel polynomial

lim N → ∞ ⁡ K n ⁡ ( q x - N ; p , N ; q ) = y n ⁡ ( q x ; p ; q ) subscript → 𝑁 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑁 𝑝 𝑁 𝑞 q-Bessel-polynomial-y 𝑛 superscript 𝑞 𝑥 𝑝 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}K_{n}\!% \left(q^{x-N};p,N;q\right)=y_{n}\!\left(q^{x};p;q\right)}}} {\displaystyle \lim_{N\rightarrow\infty}\qKrawtchouk{n}@{q^{x-N}}{p}{N}{q}=\qBesselPoly{n}@{q^x}{p}{q} }

q-Krawtchouk polynomial to q-Charlier polynomial

lim N → ∞ ⁡ K n ⁡ ( q - x ; a - 1 ⁢ q - N , N ; q ) = C n ⁡ ( q - x ; a ; q ) subscript → 𝑁 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 superscript 𝑎 1 superscript 𝑞 𝑁 𝑁 𝑞 q-Charlier-polynomial-C 𝑛 superscript 𝑞 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}K_{n}\!% \left(q^{-x};a^{-1}q^{-N},N;q\right)=C_{n}\!\left(q^{-x};a;q\right)}}} {\displaystyle \lim_{N\rightarrow\infty}\qKrawtchouk{n}@{q^{-x}}{a^{-1}q^{-N}}{N}{q}=\qCharlier{n}@{q^{-x}}{a}{q} }

q-Krawtchouk polynomial to Krawtchouk polynomial

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

Remark

K n ⁡ ( q - x ; p , N ; q ) = K x ⁡ ( λ ⁢ ( n ) ; - p ⁢ q N , N | q ) q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 dual-q-Krawtchouk-polynomial-K 𝑥 𝜆 𝑛 𝑝 superscript 𝑞 𝑁 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(q^{-x};p,N;q\right)=K_% {x}\!\left(\lambda(n);-pq^{N},N|q\right)}}} {\displaystyle \qKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\dualqKrawtchouk{x}@{\lambda(n)}{-pq^N}{N}{q} }

Substitution(s): λ ⁢ ( n ) = q - n - p ⁢ q n 𝜆 𝑛 superscript 𝑞 𝑛 𝑝 superscript 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\lambda(n)=q^{-n}-pq^{n}}}} &
λ ⁢ ( x ) = q - x + c ⁢ q x - N 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=q^{-x}+cq^{x-N}}}}


K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) = K x ⁡ ( q - n ; - c ⁢ q - N , N ; q ) dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 q-Krawtchouk-polynomial-K 𝑥 superscript 𝑞 𝑛 𝑐 superscript 𝑞 𝑁 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(\lambda(x);c,N|q\right% )=K_{x}\!\left(q^{-n};-cq^{-N},N;q\right)}}} {\displaystyle \dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q}=\qKrawtchouk{x}@{q^{-n}}{-cq^{-N}}{N}{q} }

Substitution(s): λ ⁢ ( n ) = q - n - p ⁢ q n 𝜆 𝑛 superscript 𝑞 𝑛 𝑝 superscript 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\lambda(n)=q^{-n}-pq^{n}}}} &
λ ⁢ ( x ) = q - x + c ⁢ q x - N 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=q^{-x}+cq^{x-N}}}}


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