Formula:KLS:14.15:19

<< Formula:KLS:14.15:18

formula in q-Krawtchouk

Formula:KLS:14.15:20 >>

${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{1}{1}@@{q^{-x}}{0}{q}{pqt}\,\qHyperrphis{2}{0}@@{q^{x-N},0}{-}{q}{-q^{-x}t}{}=\sum_{n=0}^{N}\frac{\left(q^{-N};q\right)_{n}}{\left(q;q\right)_{n}}q^{-\binomial{n}{2}}K_{n}\!\left(q^{-x};p,N;q\right)t^{n}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle K_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk

Bibliography

Equation in Section 14.15 of KLS.

URL links

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<< Formula:KLS:14.15:18

formula in q-Krawtchouk

Formula:KLS:14.15:20 >>