Formula:KLS:14.06:02

<< Formula:KLS:14.06:01

formula in q-Hahn

Formula:KLS:14.06:03 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\frac{\left(\alpha q,q^{-N};q\right)_{x}}{\left(q,\beta^{-1}q^{-N};q\right)_{x}}(\alpha\beta q)^{-x}Q_{m}\!\left(q^{-x};\alpha,\beta,N;q\right)Q_{n}\!\left(q^{-x};\alpha,\beta,N;q\right){}=\frac{\left(\alpha\beta q^{2};q\right)_{N}}{\left(\beta q;q\right)_{N}(\alpha q)^{N}}\frac{\left(q,\alpha\beta q^{N+2},\beta q;q\right)_{n}}{\left(\alpha q,\alpha\beta q,q^{-N};q\right)_{n}}\ \frac{(1-\alpha\beta q)(-\alpha q)^{n}}{(1-\alpha\beta q^{2n+1})}q^{\binomial{n}{2}-Nn}\,\delta_{m,n}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle Q_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:qHahn
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.6 of KLS.

URL links

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formula in q-Hahn

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