Formula:KLS:14.07:33

$\displaystyle {\displaystyle \qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}=\dualqHahn{x}@{\mu(n)}{\alpha}{\beta}{N}{q} }$

Substitution(s)

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

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

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

Proof

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Symbols List

& : logical and
$\displaystyle {\displaystyle Q_{n}}$  : $\displaystyle {\displaystyle q}$ -Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:qHahn
$\displaystyle {\displaystyle R_{n}}$  : dual $\displaystyle {\displaystyle q}$ -Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:dualqHahn