Formula:KLS:14.07:22

<< Formula:KLS:14.07:21

formula in Dual q-Hahn

Formula:KLS:14.07:23 >>

${\displaystyle{\displaystyle{\displaystyle\left(\gamma\delta qt;q\right)_{x}\cdot\qHyperrphis{2}{1}@@{q^{x-N},\gamma q^{x+1}}{\delta^{-1}q^{-N}}{q}{q^{-x}t}{}=\sum_{n=0}^{N}\frac{\left(q^{-N},\gamma q;q\right)_{n}}{\left(\delta^{-1}q^{-N},q;q\right)_{n}}R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}t^{n}}}}$

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
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle R_{n}}}}$ : dual ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:dualqHahn

Bibliography

Equation in Section 14.7 of KLS.

URL links

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formula in Dual q-Hahn

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