Formula:KLS:14.07:03

<< Formula:KLS:14.07:01

formula in Dual q-Hahn

Formula:KLS:14.07:04 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\frac{\left(\gamma q,\gamma\delta q,q^{-N};q\right)_{x}}{\left(q,\gamma\delta q^{N+2},\delta q;q\right)_{x}}\frac{(1-\gamma\delta q^{2x+1})}{(1-\gamma\delta q)(-\gamma q)^{x}}q^{Nx-\binomial{x}{2}}{}R_{m}\!\left(\mu(x);\gamma,\delta,N\right){q}R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}{}=\frac{\left(\gamma\delta q^{2};q\right)_{N}}{\left(\delta q;q\right)_{N}}(\gamma q)^{-N}\frac{\left(q,\delta^{-1}q^{-N};q\right)_{n}}{\left(\gamma q,q^{-N};q\right)_{n}}(\gamma\delta q)^{n}\,\delta_{m,n}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

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
${\displaystyle{\displaystyle{\displaystyle R_{n}}}}$ : dual ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:dualqHahn
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.7 of KLS.

URL links

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<< Formula:KLS:14.07:01

formula in Dual q-Hahn

Formula:KLS:14.07:04 >>