Formula:KLS:14.07:17

<< Formula:KLS:14.07:16

formula in Dual q-Hahn

Formula:KLS:14.07:18 >>

${\displaystyle{\displaystyle{\displaystyle\frac{\nabla\left[w(x;\gamma,\delta,N|q)R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}\right]}{\nabla\mu(x)}{}=\frac{1}{(1-q)(1-\gamma\delta)}w(x;\gamma q^{-1},\delta,N+1|q){}R_{n+1}\!\left(\mu(x);\gamma q^{-1},\delta,N+1\right){q}}}}$

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

Bibliography

Equation in Section 14.7 of KLS.

URL links

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formula in Dual q-Hahn

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