Formula:KLS:09.11:19

<< Formula:KLS:09.11:18

formula in Krawtchouk

Formula:KLS:09.11:20 >>

${\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}\sqrt{\binomial{N}{n}}K_{n}\!\left(pN+x\sqrt{2p(1-p)N};p,N\right)=\frac{\displaystyle(-1)^{n}H_{n}\left(x\right)}{\displaystyle\sqrt{2^{n}n!\left(\frac{p}{1-p}\right)^{n}}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle H_{n}}}}$ : Hermite polynomial ${\displaystyle{\displaystyle{\displaystyle H_{n}}}}$ : http://dlmf.nist.gov/18.3#T1.t1.r28

Bibliography

Equation in Section 9.11 of KLS.

URL links

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<< Formula:KLS:09.11:18

formula in Krawtchouk

Formula:KLS:09.11:20 >>