Formula:KLS:14.13:14

<< Formula:KLS:14.13:13

formula in q-Meixner

Formula:KLS:14.13:16 >>

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

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

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

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

We ask users to provide relevant URL links in this space.

<< Formula:KLS:14.13:13

formula in q-Meixner

Formula:KLS:14.13:16 >>