Formula:KLS:09.11:21

<< Formula:KLS:09.11:20

formula in Krawtchouk

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${\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{N}\binomial{N}{n}p^{n}(1-p)^{N-n}K_{n}\!\left(x;p,N\right)K_{n}\!\left(y;p,N\right)=\frac{\displaystyle\left(\frac{1-p}{p}\right)^{x}}{\dbinom{N}{x}}\delta_{x,y}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle K_{n}}}}$ : Krawtchouk polynomial : http://dlmf.nist.gov/18.19#T1.t1.r6
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 9.11 of KLS.

URL links

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