Dual q-Krawtchouk

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Dual q-Krawtchouk

Basic hypergeometric representation

K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) = \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ q - n , q - x , c ⁢ q x - N ⁢ q - N , 0 ⁢ q ⁢ q dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑁 0 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(\lambda(x);c,N|q\right% )=\qHyperrphis{3}{2}@@{q^{-n},q^{-x},cq^{x-N}}{q^{-N},0}{q}{q}}}} {\displaystyle \dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q}=\qHyperrphis{3}{2}@@{q^{-n},q^{-x},cq^{x-N}}{q^{-N},0}{q}{q} }

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


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}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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 ) = ( q x - N ; q ) n ( q - N ; q ) n ⁢ q n ⁢ x ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - n , q - x ⁢ q N - x - n + 1 ⁢ q ⁢ c ⁢ q x + 1 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 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 𝑐 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(\lambda(x);c,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}{cq^{x+1}}}}} {\displaystyle \dualqKrawtchouk{n}@{\lambda(x)}{c}{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}{cq^{x+1}} }

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}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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}}}}


Orthogonality relation(s)

∑ x = 0 N ( c ⁢ q - N , q - N ; q ) x ( q , c ⁢ q ; q ) x ⁢ ( 1 - c ⁢ q 2 ⁢ x - N ) ( 1 - c ⁢ q - N ) ⁢ c - x ⁢ q x ⁢ ( 2 ⁢ N - x ) ⁢ K m ⁡ ( λ ⁢ ( x ) ) ⁢ K n ⁡ ( λ ⁢ ( x ) ) = ( c - 1 ; q ) N ⁢ ( q ; q ) n ( q - N ; q ) n ⁢ ( c ⁢ q - N ) n ⁢ δ m , n superscript subscript 𝑥 0 𝑁 q-Pochhammer-symbol 𝑐 superscript 𝑞 𝑁 superscript 𝑞 𝑁 𝑞 𝑥 q-Pochhammer-symbol 𝑞 𝑐 𝑞 𝑞 𝑥 1 𝑐 superscript 𝑞 2 𝑥 𝑁 1 𝑐 superscript 𝑞 𝑁 superscript 𝑐 𝑥 superscript 𝑞 𝑥 2 𝑁 𝑥 dual-q-Krawtchouk-polynomial-K 𝑚 𝜆 𝑥 𝑐 𝑁 𝑞 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 q-Pochhammer-symbol superscript 𝑐 1 𝑞 𝑁 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑛 superscript 𝑐 superscript 𝑞 𝑁 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\frac{\left(cq^{-N},q^% {-N};q\right)_{x}}{\left(q,cq;q\right)_{x}}\frac{(1-cq^{2x-N})}{(1-cq^{-N})}c^% {-x}q^{x(2N-x)}K_{m}\!\left(\lambda(x)\right)K_{n}\!\left(\lambda(x)\right){}=% \left(c^{-1};q\right)_{N}\frac{\left(q;q\right)_{n}}{\left(q^{-N};q\right)_{n}% }(cq^{-N})^{n}\,\delta_{m,n}}}} {\displaystyle \sum_{x=0}^N\frac{\qPochhammer{cq^{-N},q^{-N}}{q}{x}}{\qPochhammer{q,cq}{q}{x}} \frac{(1-cq^{2x-N})}{(1-cq^{-N})}c^{-x}q^{x(2N-x)}\dualqKrawtchouk{m}@@{\lambda(x)}{c}{N}{q}\dualqKrawtchouk{n}@@{\lambda(x)}{c}{N}{q} {}=\qPochhammer{c^{-1}}{q}{N}\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q^{-N}}{q}{n}}(cq^{-N})^n\,\Kronecker{m}{n} }

Constraint(s): c < 0 𝑐 0 {\displaystyle{\displaystyle{\displaystyle c<0}}}


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


Recurrence relation

- ( 1 - q - x ) ⁢ ( 1 - c ⁢ q x - N ) ⁢ K n ⁡ ( λ ⁢ ( x ) ) = ( 1 - q n - N ) ⁢ K n + 1 ⁡ ( λ ⁢ ( x ) ) - [ ( 1 - q n - N ) + c ⁢ q - N ⁢ ( 1 - q n ) ] ⁢ K n ⁡ ( λ ⁢ ( x ) ) + c ⁢ q - N ⁢ ( 1 - q n ) ⁢ K n - 1 ⁡ ( λ ⁢ ( x ) ) 1 superscript 𝑞 𝑥 1 𝑐 superscript 𝑞 𝑥 𝑁 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 1 superscript 𝑞 𝑛 𝑁 dual-q-Krawtchouk-polynomial-K 𝑛 1 𝜆 𝑥 𝑐 𝑁 𝑞 delimited-[] 1 superscript 𝑞 𝑛 𝑁 𝑐 superscript 𝑞 𝑁 1 superscript 𝑞 𝑛 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 𝑐 superscript 𝑞 𝑁 1 superscript 𝑞 𝑛 dual-q-Krawtchouk-polynomial-K 𝑛 1 𝜆 𝑥 𝑐 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle-(1-q^{-x})(1-cq^{x-N})K_{n}\!\left(% \lambda(x)\right){}=(1-q^{n-N})K_{n+1}\!\left(\lambda(x)\right){}-\left[(1-q^{% n-N})+cq^{-N}(1-q^{n})\right]K_{n}\!\left(\lambda(x)\right){}+cq^{-N}(1-q^{n})% K_{n-1}\!\left(\lambda(x)\right)}}} {\displaystyle -(1-q^{-x})(1-cq^{x-N})\dualqKrawtchouk{n}@@{\lambda(x)}{c}{N}{q} {}=(1-q^{n-N})\dualqKrawtchouk{n+1}@@{\lambda(x)}{c}{N}{q} {}-\left[(1-q^{n-N})+cq^{-N}(1-q^n)\right]\dualqKrawtchouk{n}@@{\lambda(x)}{c}{N}{q} {}+cq^{-N}(1-q^n)\dualqKrawtchouk{n-1}@@{\lambda(x)}{c}{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}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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 ) ) := K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) assign dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(\lambda(x)\right):=K_{% n}\!\left(\lambda(x);c,N|q\right)}}} {\displaystyle \dualqKrawtchouk{n}@@{\lambda(x)}{c}{N}{q}:=\dualqKrawtchouk{n}@{\lambda(x)}{c}{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}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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}}}}


Monic recurrence relation

x ⁢ K ^ n ⁡ ( x ) = K ^ n + 1 ⁡ ( x ) + ( 1 + c ) ⁢ q n - N ⁢ K ^ n ⁡ ( x ) + c ⁢ q - N ⁢ ( 1 - q n ) ⁢ ( 1 - q n - N - 1 ) ⁢ K ^ n - 1 ⁡ ( x ) 𝑥 dual-q-Krawtchouk-polynomial-monic-p 𝑛 𝑥 𝑐 𝑁 𝑞 dual-q-Krawtchouk-polynomial-monic-p 𝑛 1 𝑥 𝑐 𝑁 𝑞 1 𝑐 superscript 𝑞 𝑛 𝑁 dual-q-Krawtchouk-polynomial-monic-p 𝑛 𝑥 𝑐 𝑁 𝑞 𝑐 superscript 𝑞 𝑁 1 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 𝑁 1 dual-q-Krawtchouk-polynomial-monic-p 𝑛 1 𝑥 𝑐 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{K}}_{n}\!\left(x\right)=% {\widehat{K}}_{n+1}\!\left(x\right)+(1+c)q^{n-N}{\widehat{K}}_{n}\!\left(x% \right){}+cq^{-N}(1-q^{n})(1-q^{n-N-1}){\widehat{K}}_{n-1}\!\left(x\right)}}} {\displaystyle x\monicdualqKrawtchouk{n}@@{x}{c}{N}{q}=\monicdualqKrawtchouk{n+1}@@{x}{c}{N}{q}+(1+c)q^{n-N}\monicdualqKrawtchouk{n}@@{x}{c}{N}{q} {}+cq^{-N}(1-q^n)(1-q^{n-N-1})\monicdualqKrawtchouk{n-1}@@{x}{c}{N}{q} }
K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) = 1 ( q - N ; q ) n ⁢ K ^ n ⁡ ( λ ⁢ ( x ) ) dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 1 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑛 dual-q-Krawtchouk-polynomial-monic-p 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(\lambda(x);c,N|q\right% )=\frac{1}{\left(q^{-N};q\right)_{n}}{\widehat{K}}_{n}\!\left(\lambda(x)\right% )}}} {\displaystyle \dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q}=\frac{1}{\qPochhammer{q^{-N}}{q}{n}}\monicdualqKrawtchouk{n}@@{\lambda(x)}{c}{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}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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}}}}


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 ) = c ⁢ q 2 ⁢ x - 2 ⁢ N - 1 ⁢ ( 1 - q x ) ⁢ ( 1 - c ⁢ q x ) ( 1 - c ⁢ q 2 ⁢ x - N - 1 ) ⁢ ( 1 - c ⁢ q 2 ⁢ x - N ) 𝐷 𝑥 𝑐 superscript 𝑞 2 𝑥 2 𝑁 1 1 superscript 𝑞 𝑥 1 𝑐 superscript 𝑞 𝑥 1 𝑐 superscript 𝑞 2 𝑥 𝑁 1 1 𝑐 superscript 𝑞 2 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle D(x)=cq^{2x-2N-1}\frac{(1-q^{x})(1-% cq^{x})}{(1-cq^{2x-N-1})(1-cq^{2x-N})}}}} &

B ⁢ ( x ) = ( 1 - q x - N ) ⁢ ( 1 - c ⁢ q x - N ) ( 1 - c ⁢ q 2 ⁢ x - N ) ⁢ ( 1 - c ⁢ q 2 ⁢ x - N + 1 ) 𝐵 𝑥 1 superscript 𝑞 𝑥 𝑁 1 𝑐 superscript 𝑞 𝑥 𝑁 1 𝑐 superscript 𝑞 2 𝑥 𝑁 1 𝑐 superscript 𝑞 2 𝑥 𝑁 1 {\displaystyle{\displaystyle{\displaystyle B(x)=\frac{(1-q^{x-N})(1-cq^{x-N})}% {(1-cq^{2x-N})(1-cq^{2x-N+1})}}}} &
y ⁢ ( x ) = K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) 𝑦 𝑥 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=K_{n}\!\left(\lambda(x);c,N|q% \right)}}} &
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}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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}}}}


Forward shift operator

K n ⁡ ( λ ⁢ ( x + 1 ) ; c , N | q ) - K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) = q - n - x ⁢ ( 1 - q n ) ⁢ ( 1 - c ⁢ q 2 ⁢ x - N + 1 ) 1 - q - N ⁢ K n - 1 ⁡ ( λ ⁢ ( x ) ; c , N - 1 | q ) dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 1 𝑐 𝑁 𝑞 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 superscript 𝑞 𝑛 𝑥 1 superscript 𝑞 𝑛 1 𝑐 superscript 𝑞 2 𝑥 𝑁 1 1 superscript 𝑞 𝑁 dual-q-Krawtchouk-polynomial-K 𝑛 1 𝜆 𝑥 𝑐 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(\lambda(x+1);c,N|q% \right)-K_{n}\!\left(\lambda(x);c,N|q\right){}=\frac{q^{-n-x}(1-q^{n})(1-cq^{2% x-N+1})}{1-q^{-N}}K_{n-1}\!\left(\lambda(x);c,N-1|q\right)}}} {\displaystyle \dualqKrawtchouk{n}@{\lambda(x+1)}{c}{N}{q}-\dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q} {}=\frac{q^{-n-x}(1-q^n)(1-cq^{2x-N+1})}{1-q^{-N}} \dualqKrawtchouk{n-1}@{\lambda(x)}{c}{N-1}{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}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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}}}}


Δ ⁢ K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) Δ ⁢ λ ⁢ ( x ) = q - n + 1 ⁢ ( 1 - q n ) ( 1 - q ) ⁢ ( 1 - q - N ) ⁢ K n - 1 ⁡ ( λ ⁢ ( x ) ; c , N - 1 | q ) Δ dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 Δ 𝜆 𝑥 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 𝑞 1 superscript 𝑞 𝑁 dual-q-Krawtchouk-polynomial-K 𝑛 1 𝜆 𝑥 𝑐 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{\Delta K_{n}\!\left(\lambda(x)% ;c,N|q\right)}{\Delta\lambda(x)}=\frac{q^{-n+1}(1-q^{n})}{(1-q)(1-q^{-N})}K_{n% -1}\!\left(\lambda(x);c,N-1|q\right)}}} {\displaystyle \frac{\Delta \dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q}}{\Delta\lambda(x)}= \frac{q^{-n+1}(1-q^n)}{(1-q)(1-q^{-N})}\dualqKrawtchouk{n-1}@{\lambda(x)}{c}{N-1}{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}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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}}}}


Backward shift operator

( 1 - q x - N - 1 ) ⁢ ( 1 - c ⁢ q x - N - 1 ) ⁢ K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) - c ⁢ q 2 ⁢ ( x - N - 1 ) ⁢ ( 1 - q x ) ⁢ ( 1 - c ⁢ q x ) ⁢ K n ⁡ ( λ ⁢ ( x - 1 ) ; c , N | q ) = q x ⁢ ( 1 - q - N - 1 ) ⁢ ( 1 - c ⁢ q 2 ⁢ x - N - 1 ) ⁢ K n + 1 ⁡ ( λ ⁢ ( x ) ; c , N + 1 | q ) 1 superscript 𝑞 𝑥 𝑁 1 1 𝑐 superscript 𝑞 𝑥 𝑁 1 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 𝑐 superscript 𝑞 2 𝑥 𝑁 1 1 superscript 𝑞 𝑥 1 𝑐 superscript 𝑞 𝑥 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 1 𝑐 𝑁 𝑞 superscript 𝑞 𝑥 1 superscript 𝑞 𝑁 1 1 𝑐 superscript 𝑞 2 𝑥 𝑁 1 dual-q-Krawtchouk-polynomial-K 𝑛 1 𝜆 𝑥 𝑐 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle(1-q^{x-N-1})(1-cq^{x-N-1})K_{n}\!% \left(\lambda(x);c,N|q\right){}-cq^{2(x-N-1)}(1-q^{x})(1-cq^{x})K_{n}\!\left(% \lambda(x-1);c,N|q\right){}=q^{x}(1-q^{-N-1})(1-cq^{2x-N-1})K_{n+1}\!\left(% \lambda(x);c,N+1|q\right)}}} {\displaystyle (1-q^{x-N-1})(1-cq^{x-N-1})\dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q} {}-cq^{2(x-N-1)}(1-q^x)(1-cq^x)\dualqKrawtchouk{n}@{\lambda(x-1)}{c}{N}{q} {}=q^x(1-q^{-N-1})(1-cq^{2x-N-1})\dualqKrawtchouk{n+1}@{\lambda(x)}{c}{N+1}{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}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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}}}}


∇ ⁡ [ w ⁢ ( x ; c , N | q ) ⁢ K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) ] ∇ ⁡ λ ⁢ ( x ) = 1 ( 1 - q ) ⁢ ( 1 - c ⁢ q - N - 1 ) ⁢ w ⁢ ( x ; c , N + 1 | q ) ⁢ K n + 1 ⁡ ( λ ⁢ ( x ) ; c , N + 1 | q ) ∇ 𝑤 𝑥 𝑐 conditional 𝑁 𝑞 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 ∇ 𝜆 𝑥 1 1 𝑞 1 𝑐 superscript 𝑞 𝑁 1 𝑤 𝑥 𝑐 𝑁 conditional 1 𝑞 dual-q-Krawtchouk-polynomial-K 𝑛 1 𝜆 𝑥 𝑐 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{\nabla\left[w(x;c,N|q)K_{n}\!% \left(\lambda(x);c,N|q\right)\right]}{\nabla\lambda(x)}{}=\frac{1}{(1-q)(1-cq^% {-N-1})}w(x;c,N+1|q)K_{n+1}\!\left(\lambda(x);c,N+1|q\right)}}} {\displaystyle \frac{\nabla\left[w(x;c,N|q)\dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q}\right]}{\nabla\lambda(x)} {}=\frac{1}{(1-q)(1-cq^{-N-1})}w(x;c,N+1|q)\dualqKrawtchouk{n+1}@{\lambda(x)}{c}{N+1}{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}}}} &
w ⁢ ( x ; c , N | q ) = ( q - N , c ⁢ q - N ; q ) x ( q , c ⁢ q ; q ) x ⁢ c - x ⁢ q 2 ⁢ N ⁢ x - x ⁢ ( x - 1 ) 𝑤 𝑥 𝑐 conditional 𝑁 𝑞 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑐 superscript 𝑞 𝑁 𝑞 𝑥 q-Pochhammer-symbol 𝑞 𝑐 𝑞 𝑞 𝑥 superscript 𝑐 𝑥 superscript 𝑞 2 𝑁 𝑥 𝑥 𝑥 1 {\displaystyle{\displaystyle{\displaystyle w(x;c,N|q)=\frac{\left(q^{-N},cq^{-% N};q\right)_{x}}{\left(q,cq;q\right)_{x}}c^{-x}q^{2Nx-x(x-1)}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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}}}}


Rodrigues-type formula

w ⁢ ( x ; c , N | q ) ⁢ K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) = ( 1 - q ) n ⁢ ( c ⁢ q - N ; q ) n ⁢ ( ∇ λ ) n ⁢ [ w ⁢ ( x ; c , N - n | q ) ] 𝑤 𝑥 𝑐 conditional 𝑁 𝑞 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 superscript 1 𝑞 𝑛 q-Pochhammer-symbol 𝑐 superscript 𝑞 𝑁 𝑞 𝑛 superscript subscript ∇ 𝜆 𝑛 delimited-[] 𝑤 𝑥 𝑐 𝑁 conditional 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle w(x;c,N|q)K_{n}\!\left(\lambda(x);c% ,N|q\right){}=(1-q)^{n}\left(cq^{-N};q\right)_{n}\left(\nabla_{\lambda}\right)% ^{n}\left[w(x;c,N-n|q)\right]}}} {\displaystyle w(x;c,N|q)\dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q} {}=(1-q)^n\qPochhammer{cq^{-N}}{q}{n}\left(\nabla_{\lambda}\right)^n\left[w(x;c,N-n|q)\right] }

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}}}} &
w ⁢ ( x ; c , N | q ) = ( q - N , c ⁢ q - N ; q ) x ( q , c ⁢ q ; q ) x ⁢ c - x ⁢ q 2 ⁢ N ⁢ x - x ⁢ ( x - 1 ) 𝑤 𝑥 𝑐 conditional 𝑁 𝑞 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑐 superscript 𝑞 𝑁 𝑞 𝑥 q-Pochhammer-symbol 𝑞 𝑐 𝑞 𝑞 𝑥 superscript 𝑐 𝑥 superscript 𝑞 2 𝑁 𝑥 𝑥 𝑥 1 {\displaystyle{\displaystyle{\displaystyle w(x;c,N|q)=\frac{\left(q^{-N},cq^{-% N};q\right)_{x}}{\left(q,cq;q\right)_{x}}c^{-x}q^{2Nx-x(x-1)}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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}}}}


∇ λ := ∇ ∇ ⁡ λ ⁢ ( x ) assign subscript ∇ 𝜆 ∇ ∇ 𝜆 𝑥 {\displaystyle{\displaystyle{\displaystyle\nabla_{\lambda}:=\frac{\nabla}{% \nabla\lambda(x)}}}} {\displaystyle \nabla_{\lambda}:=\frac{\nabla}{\nabla\lambda(x)} }

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}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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}}}}


Generating function

( c ⁢ q - N ⁢ t ; q ) x ⋅ ( q - N ⁢ t ; q ) N - x = ∑ n = 0 N ( q - N ; q ) n ( q ; q ) n ⁢ K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) ⁢ t n ⋅ q-Pochhammer-symbol 𝑐 superscript 𝑞 𝑁 𝑡 𝑞 𝑥 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑡 𝑞 𝑁 𝑥 superscript subscript 𝑛 0 𝑁 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left(cq^{-N}t;q\right)_{x}\cdot% \left(q^{-N}t;q\right)_{N-x}=\sum_{n=0}^{N}\frac{\left(q^{-N};q\right)_{n}}{% \left(q;q\right)_{n}}K_{n}\!\left(\lambda(x);c,N|q\right)t^{n}}}} {\displaystyle \qPochhammer{cq^{-N}t}{q}{x}\cdot \qPochhammer{q^{-N}t}{q}{N-x}=\sum_{n=0}^N \frac{\qPochhammer{q^{-N}}{q}{n}}{\qPochhammer{q}{q}{n}}\dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q}t^n }

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}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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}}}}


Limit relations

q-Racah polynomial to Dual q-Krawtchouk polynomial

R n ⁡ ( μ ⁢ ( x ) ; 0 , 0 , q - N - 1 , c | q ) = K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) q-Racah-polynomial-R 𝑛 𝜇 𝑥 0 0 superscript 𝑞 𝑁 1 𝑐 𝑞 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);0,0,q^{-N-1},c% \,|\,q\right)=K_{n}\!\left(\lambda(x);c,N|q\right)}}} {\displaystyle \qRacah{n}@{\mu(x)}{0}{0}{q^{-N-1}}{c}{q}=\dualqKrawtchouk{n}@{\lambda(x)}{c}{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}}}} &
μ ⁢ ( x ) = λ ⁢ ( x ) = q - x + c ⁢ q x - N 𝜇 𝑥 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\mu(x)=\lambda(x)=q^{-x}+cq^{x-N}}}} &
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}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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}}}}


Dual q-Hahn polynomial to Dual q-Krawtchouk polynomial

lim γ → 0 ⁡ R n ⁡ ( μ ⁢ ( x ) ; γ , c ⁢ γ - 1 ⁢ q - N - 1 , N ) ⁢ q = K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) subscript → 𝛾 0 dual-q-Hahn-R 𝑛 𝜇 𝑥 𝛾 𝑐 superscript 𝛾 1 superscript 𝑞 𝑁 1 𝑁 𝑞 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{\gamma\rightarrow 0}R_{n}\!% \left(\mu(x);\gamma,c\gamma^{-1}q^{-N-1},N\right){q}=K_{n}\!\left(\lambda(x);c% ,N|q\right)}}} {\displaystyle \lim_{\gamma\rightarrow 0} \dualqHahn{n}@{\mu(x)}{\gamma}{c\gamma^{-1}q^{-N-1}}{N}{q}=\dualqKrawtchouk{n}@{\lambda(x)}{c}{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}}}} &
μ ⁢ ( x ) = λ ⁢ ( x ) = q - x + c ⁢ q x - N 𝜇 𝑥 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\mu(x)=\lambda(x)=q^{-x}+cq^{x-N}}}} &
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}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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}}}}


Dual q-Krawtchouk polynomial to Al-Salam-Carlitz I polynomial

lim N → ∞ ⁡ K n ⁡ ( λ ⁢ ( x ) ; a - 1 , N | q ) = ( - 1 a ) n ⁢ q - \binomial ⁢ n ⁢ 2 ⁢ U n ( a ) ⁡ ( q x ; q ) subscript → 𝑁 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 superscript 𝑎 1 𝑁 𝑞 superscript 1 𝑎 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 superscript 𝑞 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}K_{n}\!% \left(\lambda(x);a^{-1},N|q\right)=\left(-\frac{1}{a}\right)^{n}q^{-\binomial{% n}{2}}U^{(a)}_{n}\!\left(q^{x};q\right)}}} {\displaystyle \lim_{N\rightarrow\infty}\dualqKrawtchouk{n}@{\lambda(x)}{a^{-1}}{N}{q}= \left(-\frac{1}{a}\right)^nq^{-\binomial{n}{2}}\AlSalamCarlitzI{a}{n}@{q^x}{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}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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}}}}


Dual q-Krawtchouk polynomial to Krawtchouk polynomial

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

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}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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}}}}


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}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
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}}}}


Koornwinder Addendum: Dual q-Krawtchouk

Symmetry

K n ⁡ ( x ; c , N | q ) = c n ⁢ K n ⁡ ( c - 1 ⁢ x ; c - 1 , N | q ) dual-q-Krawtchouk-polynomial-K 𝑛 𝑥 𝑐 𝑁 𝑞 superscript 𝑐 𝑛 dual-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑐 1 𝑥 superscript 𝑐 1 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(x;c,N|q\right)=c^{n}K_% {n}\!\left(c^{-1}x;c^{-1},N|q\right)}}} {\displaystyle \dualqKrawtchouk{n}@{x}{c}{N }{ q}=c^n \dualqKrawtchouk{n}@{c^{-1}x}{c^{-1}}{N }{ q} }
K n ⁡ ( x ; - 1 , N | q ) = ( - 1 ) n ⁢ K n ⁡ ( - x ; - 1 , N | q ) dual-q-Krawtchouk-polynomial-K 𝑛 𝑥 1 𝑁 𝑞 superscript 1 𝑛 dual-q-Krawtchouk-polynomial-K 𝑛 𝑥 1 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(x;-1,N|q\right)=(-1)^{% n}K_{n}\!\left(-x;-1,N|q\right)}}} {\displaystyle \dualqKrawtchouk{n}@{x}{-1}{N }{ q}=(-1)^n \dualqKrawtchouk{n}@{-x}{-1}{N }{ q} }

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