Formula:KLS:14.13:16

<< Formula:KLS:14.13:14

formula in q-Meixner

Formula:KLS:14.13:17 >>

${\displaystyle{\displaystyle{\displaystyle w(x;b,c;q)M_{n}\!\left(q^{-x};b,c;q\right)=(1-q)^{n}\left(\nabla_{q}\right)^{n}\left[w(x;bq^{n},cq^{-n};q)\right]}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle w(x;b,c;q)=\frac{\left(bq;q\right)_{x}}{\left(q,-bcq;q\right)_{x}}c^{x}q^{\binomial{x+1}{2}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle M_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Meixner polynomial : http://drmf.wmflabs.org/wiki/Definition:qMeixner
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-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

Bibliography

Equation in Section 14.13 of KLS.

URL links

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<< Formula:KLS:14.13:14

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Formula:KLS:14.13:17 >>