Formula:KLS:14.07:01

<< q-Hahn

formula in Dual q-Hahn

Formula:KLS:14.07:03 >>

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}=\qHyperrphis{3}{2}@@{q^{-n},q^{-x},\gamma\delta q^{x+1}}{\gamma q,q^{-N}}{q}{q}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots,N}}}$

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
${\displaystyle{\displaystyle{\displaystyle R_{n}}}}$ : dual ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:dualqHahn
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1

Bibliography

Equation in Section 14.7 of KLS.

URL links

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<< q-Hahn

formula in Dual q-Hahn

Formula:KLS:14.07:03 >>