Krawtchouk

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Krawtchouk

Hypergeometric representation

K n ⁡ ( x ; p , N ) = \HyperpFq ⁢ 21 ⁢ @ ⁢ @ - n , - x - N ⁢ 1 p Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 \HyperpFq 21 @ @ 𝑛 𝑥 𝑁 1 𝑝 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(x;p,N\right)=\HyperpFq% {2}{1}@@{-n,-x}{-N}{\frac{1}{p}}}}} {\displaystyle \Krawtchouk{n}@{x}{p}{N}=\HyperpFq{2}{1}@@{-n,-x}{-N}{\frac{1}{p}} }

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 \binomial ⁢ N ⁢ x ⁢ p x ⁢ ( 1 - p ) N - x ⁢ K m ⁡ ( x ; p , N ) ⁢ K n ⁡ ( x ; p , N ) = ( - 1 ) n ⁢ n ! ( - N ) n ⁢ ( 1 - p p ) n ⁢ δ m , n superscript subscript 𝑥 0 𝑁 \binomial 𝑁 𝑥 superscript 𝑝 𝑥 superscript 1 𝑝 𝑁 𝑥 Krawtchouk-polynomial-K 𝑚 𝑥 𝑝 𝑁 Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 superscript 1 𝑛 𝑛 Pochhammer-symbol 𝑁 𝑛 superscript 1 𝑝 𝑝 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\binomial{N}{x}p^{x}(1% -p)^{N-x}K_{m}\!\left(x;p,N\right)K_{n}\!\left(x;p,N\right){}=\frac{(-1)^{n}n!% }{{\left(-N\right)_{n}}}\left(\frac{1-p}{p}\right)^{n}\,\delta_{m,n}}}} {\displaystyle \sum_{x=0}^N\binomial{N}{x}p^x(1-p)^{N-x} \Krawtchouk{m}@{x}{p}{N}\Krawtchouk{n}@{x}{p}{N} {}=\frac{(-1)^nn!}{\pochhammer{-N}{n}}\left(\frac{1-p}{p}\right)^n\,\Kronecker{m}{n} }

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


Recurrence relation

- x ⁢ K n ⁡ ( x ; p , N ) = p ⁢ ( N - n ) ⁢ K n + 1 ⁡ ( x ; p , N ) - [ p ⁢ ( N - n ) + n ⁢ ( 1 - p ) ] ⁢ K n ⁡ ( x ; p , N ) + n ⁢ ( 1 - p ) ⁢ K n - 1 ⁡ ( x ; p , N ) 𝑥 Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 𝑝 𝑁 𝑛 Krawtchouk-polynomial-K 𝑛 1 𝑥 𝑝 𝑁 delimited-[] 𝑝 𝑁 𝑛 𝑛 1 𝑝 Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 𝑛 1 𝑝 Krawtchouk-polynomial-K 𝑛 1 𝑥 𝑝 𝑁 {\displaystyle{\displaystyle{\displaystyle-xK_{n}\!\left(x;p,N\right)=p(N-n)K_% {n+1}\!\left(x;p,N\right){}-\left[p(N-n)+n(1-p)\right]K_{n}\!\left(x;p,N\right% ){}+n(1-p)K_{n-1}\!\left(x;p,N\right)}}} {\displaystyle -x\Krawtchouk{n}@{x}{p}{N}=p(N-n)\Krawtchouk{n+1}@{x}{p}{N} {}-\left[p(N-n)+n(1-p)\right]\Krawtchouk{n}@{x}{p}{N} {}+n(1-p)\Krawtchouk{n-1}@{x}{p}{N} }

Monic recurrence relation

x ⁢ K ^ n ⁡ ( x ) = K ^ n + 1 ⁡ ( x ) + [ p ⁢ ( N - n ) + n ⁢ ( 1 - p ) ] ⁢ K ^ n ⁡ ( x ) + n ⁢ p ⁢ ( 1 - p ) ⁢ ( N + 1 - n ) ⁢ K ^ n - 1 ⁡ ( x ) 𝑥 Krawtchouk-polynomial-monic-p 𝑛 𝑥 𝑝 𝑁 Krawtchouk-polynomial-monic-p 𝑛 1 𝑥 𝑝 𝑁 delimited-[] 𝑝 𝑁 𝑛 𝑛 1 𝑝 Krawtchouk-polynomial-monic-p 𝑛 𝑥 𝑝 𝑁 𝑛 𝑝 1 𝑝 𝑁 1 𝑛 Krawtchouk-polynomial-monic-p 𝑛 1 𝑥 𝑝 𝑁 {\displaystyle{\displaystyle{\displaystyle x{\widehat{K}}_{n}\!\left(x\right)=% {\widehat{K}}_{n+1}\!\left(x\right)+\left[p(N-n)+n(1-p)\right]{\widehat{K}}_{n% }\!\left(x\right){}+np(1-p)(N+1-n){\widehat{K}}_{n-1}\!\left(x\right)}}} {\displaystyle x\monicKrawtchouk{n}@@{x}{p}{N}=\monicKrawtchouk{n+1}@@{x}{p}{N}+\left[p(N-n)+n(1-p)\right]\monicKrawtchouk{n}@@{x}{p}{N} {}+np(1-p)(N+1-n)\monicKrawtchouk{n-1}@@{x}{p}{N} }
K n ⁡ ( x ; p , N ) = 1 ( - N ) n ⁢ p n ⁢ K ^ n ⁡ ( x ) Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 1 Pochhammer-symbol 𝑁 𝑛 superscript 𝑝 𝑛 Krawtchouk-polynomial-monic-p 𝑛 𝑥 𝑝 𝑁 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(x;p,N\right)=\frac{1}{% {\left(-N\right)_{n}}p^{n}}{\widehat{K}}_{n}\!\left(x\right)}}} {\displaystyle \Krawtchouk{n}@{x}{p}{N}=\frac{1}{\pochhammer{-N}{n}p^n}\monicKrawtchouk{n}@@{x}{p}{N} }

Difference equation

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

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


Forward shift operator

K n ⁡ ( x + 1 ; p , N ) - K n ⁡ ( x ; p , N ) = - n N ⁢ p ⁢ K n - 1 ⁡ ( x ; p , N - 1 ) Krawtchouk-polynomial-K 𝑛 𝑥 1 𝑝 𝑁 Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 𝑛 𝑁 𝑝 Krawtchouk-polynomial-K 𝑛 1 𝑥 𝑝 𝑁 1 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(x+1;p,N\right)-K_{n}\!% \left(x;p,N\right)=-\frac{n}{Np}K_{n-1}\!\left(x;p,N-1\right)}}} {\displaystyle \Krawtchouk{n}@{x+1}{p}{N}-\Krawtchouk{n}@{x}{p}{N}=-\frac{n}{Np}\Krawtchouk{n-1}@{x}{p}{N-1} }
Δ ⁢ K n ⁡ ( x ; p , N ) = - n N ⁢ p ⁢ K n - 1 ⁡ ( x ; p , N - 1 ) Δ Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 𝑛 𝑁 𝑝 Krawtchouk-polynomial-K 𝑛 1 𝑥 𝑝 𝑁 1 {\displaystyle{\displaystyle{\displaystyle\Delta K_{n}\!\left(x;p,N\right)=-% \frac{n}{Np}K_{n-1}\!\left(x;p,N-1\right)}}} {\displaystyle \Delta \Krawtchouk{n}@{x}{p}{N}=-\frac{n}{Np}\Krawtchouk{n-1}@{x}{p}{N-1} }

Backward shift operator

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

Rodrigues-type formula

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

Generating functions

( 1 - ( 1 - p ) p ⁢ t ) x ⁢ ( 1 + t ) N - x = ∑ n = 0 N \binomial ⁢ N ⁢ n ⁢ K n ⁡ ( x ; p , N ) ⁢ t n superscript 1 1 𝑝 𝑝 𝑡 𝑥 superscript 1 𝑡 𝑁 𝑥 superscript subscript 𝑛 0 𝑁 \binomial 𝑁 𝑛 Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left(1-\frac{(1-p)}{p}t\right)^{x}(% 1+t)^{N-x}=\sum_{n=0}^{N}\binomial{N}{n}K_{n}\!\left(x;p,N\right)t^{n}}}} {\displaystyle \left(1-\frac{(1-p)}{p}t\right)^x(1+t)^{N-x}= \sum_{n=0}^N\binomial{N}{n}\Krawtchouk{n}@{x}{p}{N}t^n }
[ e t ⁢ \HyperpFq ⁢ 11 ⁢ @ ⁢ @ - x - N - t p ] N = ∑ n = 0 N K n ⁡ ( x ; p , N ) n ! ⁢ t n subscript delimited-[] 𝑡 \HyperpFq 11 @ @ 𝑥 𝑁 𝑡 𝑝 𝑁 superscript subscript 𝑛 0 𝑁 Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left[{\mathrm{e}^{t}}\,\HyperpFq{1}% {1}@@{-x}{-N}{-\frac{t}{p}}\right]_{N}=\sum_{n=0}^{N}\frac{K_{n}\!\left(x;p,N% \right)}{n!}t^{n}}}} {\displaystyle \left[\expe^t\,\HyperpFq{1}{1}@@{-x}{-N}{-\frac{t}{p}}\right]_N= \sum_{n=0}^N\frac{\Krawtchouk{n}@{x}{p}{N}}{n!}t^n }
[ ( 1 - t ) - γ ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ γ , - x - N ⁢ t p ⁢ ( t - 1 ) ] N = ∑ n = 0 N ( γ ) n n ! ⁢ K n ⁡ ( x ; p , N ) ⁢ t n subscript superscript 1 𝑡 𝛾 \HyperpFq 21 @ @ 𝛾 𝑥 𝑁 𝑡 𝑝 𝑡 1 𝑁 superscript subscript 𝑛 0 𝑁 Pochhammer-symbol 𝛾 𝑛 𝑛 Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left[(1-t)^{-\gamma}\,\HyperpFq{2}{% 1}@@{\gamma,-x}{-N}{\frac{t}{p(t-1)}}\right]_{N}{}=\sum_{n=0}^{N}\frac{{\left(% \gamma\right)_{n}}}{n!}K_{n}\!\left(x;p,N\right)t^{n}}}} {\displaystyle \left[(1-t)^{-\gamma}\,\HyperpFq{2}{1}@@{\gamma,-x}{-N}{\frac{t}{p(t-1)}}\right]_N {}=\sum_{n=0}^N\frac{\pochhammer{\gamma}{n}}{n!}\Krawtchouk{n}@{x}{p}{N}t^n }

Constraint(s): γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


Limit relations

Hahn polynomial to Krawtchouk polynomial

lim t → ∞ ⁡ Q n ⁡ ( x ; p ⁢ t , ( 1 - p ) ⁢ t , N ) = K n ⁡ ( x ; p , N ) subscript → 𝑡 Hahn-polynomial-Q 𝑛 𝑥 𝑝 𝑡 1 𝑝 𝑡 𝑁 Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 {\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}Q_{n}\!% \left(x;pt,(1-p)t,N\right)=K_{n}\!\left(x;p,N\right)}}} {\displaystyle \lim_{t\rightarrow\infty}\Hahn{n}@{x}{pt}{(1-p)t}{N}=\Krawtchouk{n}@{x}{p}{N} }

Dual Hahn polynomial to Krawtchouk polynomial

lim t → ∞ ⁡ R n ⁡ ( λ ⁢ ( x ) ; p ⁢ t , ( 1 - p ) ⁢ t , N ) = K n ⁡ ( x ; p , N ) subscript → 𝑡 dual-Hahn-R 𝑛 𝜆 𝑥 𝑝 𝑡 1 𝑝 𝑡 𝑁 Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 {\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}R_{n}\!% \left(\lambda(x);pt,(1-p)t,N\right)=K_{n}\!\left(x;p,N\right)}}} {\displaystyle \lim_{t\rightarrow\infty}\dualHahn{n}@{\lambda(x)}{pt}{(1-p)t}{N}=\Krawtchouk{n}@{x}{p}{N} }

Krawtchouk polynomial to Charlier polynomial

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

Krawtchouk polynomial to Hermite polynomial

lim N → ∞ ⁡ \binomial ⁢ N ⁢ n ⁢ K n ⁡ ( p ⁢ N + x ⁢ 2 ⁢ p ⁢ ( 1 - p ) ⁢ N ; p , N ) = ( - 1 ) n ⁢ H n ⁡ ( x ) 2 n ⁢ n ! ⁢ ( p 1 - p ) n subscript → 𝑁 \binomial 𝑁 𝑛 Krawtchouk-polynomial-K 𝑛 𝑝 𝑁 𝑥 2 𝑝 1 𝑝 𝑁 𝑝 𝑁 superscript 1 𝑛 Hermite-polynomial-H 𝑛 𝑥 superscript 2 𝑛 𝑛 superscript 𝑝 1 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}\sqrt{% \binomial{N}{n}}K_{n}\!\left(pN+x\sqrt{2p(1-p)N};p,N\right)=\frac{% \displaystyle(-1)^{n}H_{n}\left(x\right)}{\displaystyle\sqrt{2^{n}n!\left(% \frac{p}{1-p}\right)^{n}}}}}} {\displaystyle \lim_{N\rightarrow\infty} \sqrt{\binomial{N}{n}}\Krawtchouk{n}@{pN+x\sqrt{2p(1-p)N}}{p}{N} =\frac{\displaystyle (-1)^n\Hermite{n}@{x}}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}} }

Remarks

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

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


∑ n = 0 N \binomial ⁢ N ⁢ n ⁢ p n ⁢ ( 1 - p ) N - n ⁢ K n ⁡ ( x ; p , N ) ⁢ K n ⁡ ( y ; p , N ) = ( 1 - p p ) x ( N x ) ⁢ δ x , y superscript subscript 𝑛 0 𝑁 \binomial 𝑁 𝑛 superscript 𝑝 𝑛 superscript 1 𝑝 𝑁 𝑛 Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 Krawtchouk-polynomial-K 𝑛 𝑦 𝑝 𝑁 superscript 1 𝑝 𝑝 𝑥 binomial 𝑁 𝑥 Kronecker-delta 𝑥 𝑦 {\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{N}\binomial{N}{n}p^{n}(1% -p)^{N-n}K_{n}\!\left(x;p,N\right)K_{n}\!\left(y;p,N\right)=\frac{% \displaystyle\left(\frac{1-p}{p}\right)^{x}}{\dbinom{N}{x}}\delta_{x,y}}}} {\displaystyle \sum_{n=0}^N\binomial{N}{n}p^n(1-p)^{N-n} \Krawtchouk{n}@{x}{p}{N}\Krawtchouk{n}@{y}{p}{N}= \frac{\displaystyle\left(\frac{1-p}{p}\right)^x}{\dbinom{N}{x}}\Kronecker{x}{y} }
K n ⁡ ( x ; p , N ) = M n ⁡ ( x ; - N , ( p - 1 ) - 1 ⁢ p ) Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 Meixner-polynomial-M 𝑛 𝑥 𝑁 superscript 𝑝 1 1 𝑝 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(x;p,N\right)=M_{n}\!% \left(x;-N,(p-1)^{-1}p\right)}}} {\displaystyle \Krawtchouk{n}@{x}{p}{N}=\Meixner{n}@{x}{-N}{(p-1)^{-1}p} }

Koornwinder Addendum: Krawtchouk

Krawtchouk: Special values

K n ⁡ ( 0 ; p , N ) = 1 , K n ⁡ ( N ; p , N ) Krawtchouk-polynomial-K 𝑛 0 𝑝 𝑁 1 Krawtchouk-polynomial-K 𝑛 𝑁 𝑝 𝑁 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(0;p,N\right)=1,\qquad K% _{n}\!\left(N;p,N\right)}}} {\displaystyle \Krawtchouk{n}@{0}{p}{N}=1,\qquad \Krawtchouk{n}@{N}{p}{N} }
K n ⁡ ( 0 ; p , N ) = ( 1 - p - 1 ) n Krawtchouk-polynomial-K 𝑛 0 𝑝 𝑁 superscript 1 superscript 𝑝 1 𝑛 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(0;p,N\right)=(1-p^{-1}% )^{n}}}} {\displaystyle \Krawtchouk{n}@{0}{p}{N} =(1-p^{-1})^n }
K n ⁡ ( x ; p , N ) = K x ⁡ ( n ; p , N )    ( n , x ∈ { 0 , 1 , … , N } ) fragments Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 Krawtchouk-polynomial-K 𝑥 𝑛 𝑝 𝑁 italic-   fragments ( n , x fragments { 0 , 1 , … , N } ) {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(x;p,N\right)=K_{x}\!% \left(n;p,N\right)\qquad(n,x\in\{0,1,\ldots,N\})}}} {\displaystyle \Krawtchouk{n}@{x}{p}{N}=\Krawtchouk{x}@{n}{p}{N}\qquad (n,x\in \{0,1,\ldots,N\}) }
K N ⁡ ( x ; p , N ) = ( 1 - p - 1 ) x    ( x ∈ { 0 , 1 , … , N } ) fragments Krawtchouk-polynomial-K 𝑁 𝑥 𝑝 𝑁 superscript fragments ( 1 superscript 𝑝 1 ) 𝑥 italic-   fragments ( x fragments { 0 , 1 , … , N } ) {\displaystyle{\displaystyle{\displaystyle K_{N}\!\left(x;p,N\right)=(1-p^{-1}% )^{x}\qquad(x\in\{0,1,\ldots,N\})}}} {\displaystyle \Krawtchouk{N}@{x}{p}{N}=(1-p^{-1})^x\qquad(x\in\{0,1,\ldots,N\}) }

Symmetry

K n ⁡ ( N - x ; p , N ) K n ⁡ ( N ; p , N ) = K n ⁡ ( x ; 1 - p , N ) Krawtchouk-polynomial-K 𝑛 𝑁 𝑥 𝑝 𝑁 Krawtchouk-polynomial-K 𝑛 𝑁 𝑝 𝑁 Krawtchouk-polynomial-K 𝑛 𝑥 1 𝑝 𝑁 {\displaystyle{\displaystyle{\displaystyle\frac{K_{n}\!\left(N-x;p,N\right)}{K% _{n}\!\left(N;p,N\right)}=K_{n}\!\left(x;1-p,N\right)}}} {\displaystyle \frac{\Krawtchouk{n}@{N-x}{p}{N}}{\Krawtchouk{n}@{N}{p}{N}}=\Krawtchouk{n}@{x}{1-p}{N} }
K N - n ⁡ ( x ; p , N ) K N ⁡ ( x ; p , N ) = K n ⁡ ( x ; 1 - p , N )    ( n , x ∈ { 0 , 1 , … , N } ) fragments Krawtchouk-polynomial-K 𝑁 𝑛 𝑥 𝑝 𝑁 Krawtchouk-polynomial-K 𝑁 𝑥 𝑝 𝑁 Krawtchouk-polynomial-K 𝑛 𝑥 1 𝑝 𝑁 italic-   fragments ( n , x fragments { 0 , 1 , … , N } ) {\displaystyle{\displaystyle{\displaystyle\frac{K_{N-n}\!\left(x;p,N\right)}{K% _{N}\!\left(x;p,N\right)}=K_{n}\!\left(x;1-p,N\right)\qquad(n,x\in\{0,1,\ldots% ,N\})}}} {\displaystyle \frac{\Krawtchouk{N-n}@{x}{p}{N}}{\Krawtchouk{N}@{x}{p}{N}}=\Krawtchouk{n}@{x}{1-p}{N} \qquad(n,x\in\{0,1,\ldots,N\}) }
K N - n ⁡ ( N - x ; p , N ) = ( p p - 1 ) n + x - N K n ⁡ ( x ; p , N )    ( n , x ∈ { 0 , 1 , … , N } ) fragments Krawtchouk-polynomial-K 𝑁 𝑛 𝑁 𝑥 𝑝 𝑁 superscript fragments ( 𝑝 𝑝 1 ) 𝑛 𝑥 𝑁 Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 italic-   fragments ( n , x fragments { 0 , 1 , … , N } ) {\displaystyle{\displaystyle{\displaystyle K_{N-n}\!\left(N-x;p,N\right)=\left% (\frac{p}{p-1}\right)^{n+x-N}K_{n}\!\left(x;p,N\right)\qquad(n,x\in\{0,1,% \ldots,N\})}}} {\displaystyle \Krawtchouk{N-n}@{N-x}{p}{N}=\left(\frac p{p-1}\right)^{n+x-N}\Krawtchouk{n}@{x}{p}{N} \qquad(n,x\in\{0,1,\ldots,N\}) }
K n ⁡ ( N - x ; 1 2 , N ) = ( - 1 ) n ⁢ K n ⁡ ( x ; 1 2 , N ) Krawtchouk-polynomial-K 𝑛 𝑁 𝑥 1 2 𝑁 superscript 1 𝑛 Krawtchouk-polynomial-K 𝑛 𝑥 1 2 𝑁 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(N-x;\frac{1}{2},N% \right)=(-1)^{n}K_{n}\!\left(x;\frac{1}{2},N\right)}}} {\displaystyle \Krawtchouk{n}@{N-x}{\frac12}{N}=(-1)^n \Krawtchouk{n}@{x}{\frac12}{N} }
K 2 ⁢ m + 1 ⁡ ( N ; 1 2 , 2 ⁢ N ) = 0 Krawtchouk-polynomial-K 2 𝑚 1 𝑁 1 2 2 𝑁 0 {\displaystyle{\displaystyle{\displaystyle K_{2m+1}\!\left(N;\frac{1}{2},2N% \right)=0}}} {\displaystyle \Krawtchouk{2m+1}@{N}{\frac12}{2N}=0 }
K 2 ⁢ m ⁡ ( N ; 1 2 , 2 ⁢ N ) = ( 1 2 ) m ( - N + 1 2 ) m Krawtchouk-polynomial-K 2 𝑚 𝑁 1 2 2 𝑁 Pochhammer-symbol 1 2 𝑚 Pochhammer-symbol 𝑁 1 2 𝑚 {\displaystyle{\displaystyle{\displaystyle K_{2m}\!\left(N;\frac{1}{2},2N% \right)=\frac{{\left(\frac{1}{2}\right)_{m}}}{{\left(-N+\frac{1}{2}\right)_{m}% }}}}} {\displaystyle \Krawtchouk{2m}@{N}{\frac12}{2N}=\frac{\pochhammer{\frac12}{m}}{\pochhammer{-N+\frac12}{m}} }

Quadratic transformations

K 2 ⁢ m ⁡ ( x + N ; 1 2 , 2 ⁢ N ) = ( 1 2 ) m ( - N + 1 2 ) m ⁢ R m ⁡ ( x 2 ; - 1 2 , - 1 2 , N ) Krawtchouk-polynomial-K 2 𝑚 𝑥 𝑁 1 2 2 𝑁 Pochhammer-symbol 1 2 𝑚 Pochhammer-symbol 𝑁 1 2 𝑚 dual-Hahn-R 𝑚 superscript 𝑥 2 1 2 1 2 𝑁 {\displaystyle{\displaystyle{\displaystyle K_{2m}\!\left(x+N;\frac{1}{2},2N% \right)=\frac{{\left(\frac{1}{2}\right)_{m}}}{{\left(-N+\frac{1}{2}\right)_{m}% }}R_{m}\!\left(x^{2};-\frac{1}{2},-\frac{1}{2},N\right)}}} {\displaystyle \Krawtchouk{2m}@{x+N}{\frac12}{2N}=\frac{\pochhammer{\frac12}{m}}{\pochhammer{-N+\frac12}{m}} \dualHahn{m}@{x^2}{-\frac12}{-\frac12}{N} }
K 2 ⁢ m + 1 ⁡ ( x + N ; 1 2 , 2 ⁢ N ) = - ( 3 2 ) m N ⁢ ( - N + 1 2 ) m ⁢ x ⁢ R m ⁡ ( x 2 - 1 ; 1 2 , 1 2 , N - 1 ) Krawtchouk-polynomial-K 2 𝑚 1 𝑥 𝑁 1 2 2 𝑁 Pochhammer-symbol 3 2 𝑚 𝑁 Pochhammer-symbol 𝑁 1 2 𝑚 𝑥 dual-Hahn-R 𝑚 superscript 𝑥 2 1 1 2 1 2 𝑁 1 {\displaystyle{\displaystyle{\displaystyle K_{2m+1}\!\left(x+N;\frac{1}{2},2N% \right)=-\frac{{\left(\tfrac{3}{2}\right)_{m}}}{N{\left(-N+\frac{1}{2}\right)_% {m}}}xR_{m}\!\left(x^{2}-1;\frac{1}{2},\frac{1}{2},N-1\right)}}} {\displaystyle \Krawtchouk{2m+1}@{x+N}{\frac12}{2N}=- \frac{\pochhammer{\tfrac32}{m}}{N \pochhammer{-N+\frac12}{m}} x \dualHahn{m}@{x^2-1}{\frac12}{\frac12}{N-1} }
K 2 ⁢ m ⁡ ( x + N + 1 ; 1 2 , 2 ⁢ N + 1 ) = ( 1 2 ) m ( - N - 1 2 ) m ⁢ R m ⁡ ( x ⁢ ( x + 1 ) ; - 1 2 , 1 2 , N ) Krawtchouk-polynomial-K 2 𝑚 𝑥 𝑁 1 1 2 2 𝑁 1 Pochhammer-symbol 1 2 𝑚 Pochhammer-symbol 𝑁 1 2 𝑚 dual-Hahn-R 𝑚 𝑥 𝑥 1 1 2 1 2 𝑁 {\displaystyle{\displaystyle{\displaystyle K_{2m}\!\left(x+N+1;\frac{1}{2},2N+% 1\right)=\frac{{\left(\tfrac{1}{2}\right)_{m}}}{{\left(-N-\frac{1}{2}\right)_{% m}}}R_{m}\!\left(x(x+1);-\frac{1}{2},\frac{1}{2},N\right)}}} {\displaystyle \Krawtchouk{2m}@{x+N+1}{\frac12}{2N+1}=\frac{\pochhammer{\tfrac12}{m}}{\pochhammer{-N-\frac12}{m}} \dualHahn{m}@{x(x+1)}{-\frac12}{\frac12}{N} }
K 2 ⁢ m + 1 ⁡ ( x + N + 1 ; 1 2 , 2 ⁢ N + 1 ) = ( 3 2 ) m ( - N - 1 2 ) m + 1 ⁢ ( x + 1 2 ) ⁢ R m ⁡ ( x ⁢ ( x + 1 ) ; 1 2 , - 1 2 , N ) Krawtchouk-polynomial-K 2 𝑚 1 𝑥 𝑁 1 1 2 2 𝑁 1 Pochhammer-symbol 3 2 𝑚 Pochhammer-symbol 𝑁 1 2 𝑚 1 𝑥 1 2 dual-Hahn-R 𝑚 𝑥 𝑥 1 1 2 1 2 𝑁 {\displaystyle{\displaystyle{\displaystyle K_{2m+1}\!\left(x+N+1;\frac{1}{2},2% N+1\right)=\frac{{\left(\tfrac{3}{2}\right)_{m}}}{{\left(-N-\frac{1}{2}\right)% _{m+1}}}(x+\frac{1}{2})R_{m}\!\left(x(x+1);\frac{1}{2},-\frac{1}{2},N\right)}}} {\displaystyle \Krawtchouk{2m+1}@{x+N+1}{\frac12}{2N+1}=\frac{\pochhammer{\tfrac32}{m}}{\pochhammer{-N-\frac12}{m+1}} (x+\frac12) \dualHahn{m}@{x(x+1)}{\frac12}{-\frac12}{N} }

Generating functions

∑ x = 0 N ( N x ) ⁢ K m ⁡ ( x ; p , N ) ⁢ K n ⁡ ( x ; q , N ) ⁢ z x = ( p - z + p ⁢ z p ) m ⁢ ( q - z + q ⁢ z q ) n ⁢ ( 1 + z ) N - m - n ⁢ K m ⁡ ( n ; - ( p - z + p ⁢ z ) ⁢ ( q - z + q ⁢ z ) z , N ) superscript subscript 𝑥 0 𝑁 binomial 𝑁 𝑥 Krawtchouk-polynomial-K 𝑚 𝑥 𝑝 𝑁 Krawtchouk-polynomial-K 𝑛 𝑥 𝑞 𝑁 superscript 𝑧 𝑥 superscript 𝑝 𝑧 𝑝 𝑧 𝑝 𝑚 superscript 𝑞 𝑧 𝑞 𝑧 𝑞 𝑛 superscript 1 𝑧 𝑁 𝑚 𝑛 Krawtchouk-polynomial-K 𝑚 𝑛 𝑝 𝑧 𝑝 𝑧 𝑞 𝑧 𝑞 𝑧 𝑧 𝑁 {\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\genfrac{(}{)}{0.0pt}{% }{N}{x}K_{m}\!\left(x;p,N\right)K_{n}\!\left(x;q,N\right)z^{x}=\left(\frac{p-z% +pz}{p}\right)^{m}\left(\frac{q-z+qz}{q}\right)^{n}(1+z)^{N-m-n}K_{m}\!\left(n% ;-\frac{(p-z+pz)(q-z+qz)}{z},N\right)}}} {\displaystyle \sum_{x=0}^N\binom Nx \Krawtchouk{m}@{x}{p}{N}\Krawtchouk{n}@{x}{q}{N}z^x =\left(\frac{p-z+pz}p\right)^m \left(\frac{q-z+qz}q\right)^n (1+z)^{N-m-n} \Krawtchouk{m}@{n}{- \frac{(p-z+pz)(q-z+qz)}z}{N} }

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