Formula:KLS:14.15:17: Difference between revisions

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Latest revision as of 08:38, 22 December 2019


w ( x ; p , N ; q ) K n ( q - x ; p , N ; q ) = ( 1 - q ) n ( q ) n [ w ( x ; p q 2 n , N - n ; q ) ] 𝑤 𝑥 𝑝 𝑁 𝑞 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 superscript 1 𝑞 𝑛 superscript subscript 𝑞 𝑛 delimited-[] 𝑤 𝑥 𝑝 superscript 𝑞 2 𝑛 𝑁 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle w(x;p,N;q)K_{n}\!\left(q^{-x};p,N;q% \right)=(1-q)^{n}\left(\nabla_{q}\right)^{n}\left[w(x;pq^{2n},N-n;q)\right]}}}

Substitution(s)

w ( x ; p , N ; q ) = ( q - N ; q ) x ( q ; q ) x ( - q p ) x 𝑤 𝑥 𝑝 𝑁 𝑞 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑥 q-Pochhammer-symbol 𝑞 𝑞 𝑥 superscript 𝑞 𝑝 𝑥 {\displaystyle{\displaystyle{\displaystyle w(x;p,N;q)=\frac{\left(q^{-N};q% \right)_{x}}{\left(q;q\right)_{x}}\left(-\frac{q}{p}\right)^{x}}}}


Proof

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Symbols List

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
( 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 14.15 of KLS.

URL links

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