Formula:KLS:14.13:02

<< Formula:KLS:14.13:01

formula in q-Meixner

Formula:KLS:14.13:03 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{\infty}\frac{\left(bq;q\right)_{x}}{\left(q,-bcq;q\right)_{x}}c^{x}q^{\binomial{x}{2}}M_{m}\!\left(q^{-x};b,c;q\right)M_{n}\!\left(q^{-x};b,c;q\right){}=\frac{\left(-c;q\right)_{\infty}}{\left(-bcq;q\right)_{\infty}}\frac{\left(q,-c^{-1}q;q\right)_{n}}{\left(bq;q\right)_{n}}q^{-n}\,\delta_{m,n}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle 0\leq bq<1}}}$ &
${\displaystyle{\displaystyle{\displaystyle c>0}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle M_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Meixner polynomial : http://drmf.wmflabs.org/wiki/Definition:qMeixner
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.13 of KLS.

URL links

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