Formula:KLS:14.16:17

<< Formula:KLS:14.16:15

formula in Affine q-Krawtchouk

Formula:KLS:14.16:18 >>

${\displaystyle{\displaystyle{\displaystyle w(x;p,N;q)K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right){}=\frac{(-1)^{n}q^{-Nn+\binomial{n}{2}}(1-q)^{n}}{\left(q^{-N};q\right)_{n}}\left(\nabla_{q}\right)^{n}\left[w(x;pq^{n},N-n;q)\right]}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle w(x;p,N;q)=\frac{\left(pq;q\right)_{x}}{\left(q;q\right)_{x}\left(q;q\right)_{N-x}}p^{-x}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle K^{\mathrm{Aff}}_{n}}}}$ : affine ${\displaystyle{\displaystyle{\displaystyle q}}}$-Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1

Bibliography

Equation in Section 14.16 of KLS.

URL links

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<< Formula:KLS:14.16:15

formula in Affine q-Krawtchouk

Formula:KLS:14.16:18 >>