Formula:KLS:14.16:02

<< Formula:KLS:14.16:01

formula in Affine q-Krawtchouk

Formula:KLS:14.16:03 >>

${\displaystyle{\displaystyle{\displaystyle K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)=\frac{(-pq)^{n}q^{\binomial{n}{2}}}{\left(pq;q\right)_{n}}\ \qHyperrphis{2}{1}@@{q^{-n},q^{x-N}}{q^{-N}}{q}{\frac{q^{-x}}{p}}}}}$

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:14.16:01

formula in Affine q-Krawtchouk

Formula:KLS:14.16:03 >>