Formula:KLS:09.11:37

<< Formula:KLS:09.11:36

formula in Krawtchouk

Formula:KLS:09.12:01 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\genfrac{(}{)}{0.0pt}{}{N}{x}K_{m}\!\left(x;p,N\right)K_{n}\!\left(x;q,N\right)z^{x}=\left(\frac{p-z+pz}{p}\right)^{m}\left(\frac{q-z+qz}{q}\right)^{n}(1+z)^{N-m-n}K_{m}\!\left(n;-\frac{(p-z+pz)(q-z+qz)}{z},N\right)}}}$

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

Bibliography

Equation in Section 9.11 of KLS.

URL links

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<< Formula:KLS:09.11:36

formula in Krawtchouk

Formula:KLS:09.12:01 >>