Formula:KLS:14.17:26

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lim q → 1 ⁡ K n ⁡ ( λ ⁢ ( x ) ; 1 - p - 1 , N | q ) = K n ⁡ ( x ; p , N ) subscript → 𝑞 1 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 1 superscript 𝑝 1 𝑁 𝑞 Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}K_{n}\!\left(% \lambda(x);1-p^{-1},N|q\right)=K_{n}\!\left(x;p,N\right)}}}

Substitution(s)

λ ⁢ ( n ) = q - n - p ⁢ q n 𝜆 𝑛 superscript 𝑞 𝑛 𝑝 superscript 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\lambda(n)=q^{-n}-pq^{n}}}} &

λ ⁢ ( x ) = q - x + c ⁢ q x - N 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=q^{-x}+cq^{x-N}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
λ ⁢ ( n ) = q - n - p ⁢ q n 𝜆 𝑛 superscript 𝑞 𝑛 𝑝 superscript 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\lambda(n)=q^{-n}-pq^{n}}}} &

λ ⁢ ( x ) = q - x + c ⁢ q x - N 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=q^{-x}+cq^{x-N}}}}


Proof

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Symbols List

& : logical and
K n subscript 𝐾 𝑛 {\displaystyle{\displaystyle{\displaystyle K_{n}}}}  : dual q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk
K n subscript 𝐾 𝑛 {\displaystyle{\displaystyle{\displaystyle K_{n}}}}  : Krawtchouk polynomial : http://dlmf.nist.gov/18.19#T1.t1.r6

Bibliography

Equation in Section 14.17 of KLS.

URL links

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