Formula:KLS:14.17:27

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K n ⁡ ( q - x ; p , N ; q ) = K x ⁡ ( λ ⁢ ( n ) ; - p ⁢ q N , N | q ) q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 dual-q-Krawtchouk-polynomial-K 𝑥 𝜆 𝑛 𝑝 superscript 𝑞 𝑁 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(q^{-x};p,N;q\right)=K_% {x}\!\left(\lambda(n);-pq^{N},N|q\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}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk
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

Bibliography

Equation in Section 14.17 of KLS.

URL links

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