Formula:KLS:09.15:22

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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}}}}}}

Proof

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Symbols List

( n k ) binomial 𝑛 𝑘 {\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
K n subscript 𝐾 𝑛 {\displaystyle{\displaystyle{\displaystyle K_{n}}}}  : Krawtchouk polynomial : http://dlmf.nist.gov/18.19#T1.t1.r6
H n subscript 𝐻 𝑛 {\displaystyle{\displaystyle{\displaystyle H_{n}}}}  : Hermite polynomial H n subscript 𝐻 𝑛 {\displaystyle{\displaystyle{\displaystyle H_{n}}}}  : http://dlmf.nist.gov/18.3#T1.t1.r28

Bibliography

Equation in Section 9.15 of KLS.

URL links

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