Formula:KLS:14.24:01

<< q-Charlier

formula in Al-Salam-Carlitz I

Formula:KLS:14.24:02 >>

${\displaystyle{\displaystyle{\displaystyle U^{(a)}_{n}\!\left(x;q\right)=(-a)^{n}q^{\binomial{n}{2}}\,\qHyperrphis{2}{1}@@{q^{-n},x^{-1}}{0}{q}{\frac{qx}{a}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle U^{(n)}_{\alpha}}}}$ : Al-Salam-Carlitz I polynomial : http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1

Bibliography

Equation in Section 14.24 of KLS.

URL links

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<< q-Charlier

formula in Al-Salam-Carlitz I

Formula:KLS:14.24:02 >>