Formula:KLS:14.16:20

<< Formula:KLS:14.16:19

formula in Affine q-Krawtchouk

Formula:KLS:14.16:21 >>

${\displaystyle{\displaystyle{\displaystyle\left(-pq^{-N+1}t;q\right)_{x}\cdot\qHyperrphis{2}{0}@@{q^{x-N},pq^{x+1}}{-}{q}{-q^{-x}t}{}=\sum_{n=0}^{N}\frac{\left(pq,q^{-N};q\right)_{n}}{\left(q;q\right)_{n}}q^{-\binomial{n}{2}}K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)t^{n}}}}$

Proof

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Symbols List

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

Bibliography

Equation in Section 14.16 of KLS.

URL links

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<< Formula:KLS:14.16:19

formula in Affine q-Krawtchouk

Formula:KLS:14.16:21 >>