Formula:KLS:14.17:02

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K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) = ( q x - N ; q ) n ( q - N ; q ) n ⁢ q n ⁢ x ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - n , q - x ⁢ q N - x - n + 1 ⁢ q ⁢ c ⁢ q x + 1 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 q-Pochhammer-symbol superscript 𝑞 𝑥 𝑁 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑛 superscript 𝑞 𝑛 𝑥 \qHyperrphis 21 @ @ superscript 𝑞 𝑛 superscript 𝑞 𝑥 superscript 𝑞 𝑁 𝑥 𝑛 1 𝑞 𝑐 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(\lambda(x);c,N|q\right% )=\frac{\left(q^{x-N};q\right)_{n}}{\left(q^{-N};q\right)_{n}q^{nx}}\ % \qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{q^{N-x-n+1}}{q}{cq^{x+1}}}}}

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
( a ; q ) n subscript 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1

Bibliography

Equation in Section 14.17 of KLS.

URL links

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