Formula:KLS:14.07:10

<< Formula:KLS:14.07:09

formula in Dual q-Hahn

Formula:KLS:14.07:14 >>

${\displaystyle{\displaystyle{\displaystyle q^{-n}(1-q^{n})y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle D(x)=-\frac{\gamma q^{x-N}(1-q^{x})(1-\gamma\delta q^{x+N+1})(1-\delta q^{x})}{(1-\gamma\delta q^{2x})(1-\gamma\delta q^{2x+1})}}}}$ &

${\displaystyle{\displaystyle{\displaystyle B(x)=\frac{(1-q^{x-N})(1-\gamma q^{x+1})(1-\gamma\delta q^{x+1})}{(1-\gamma\delta q^{2x+1})(1-\gamma\delta q^{2x+2})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle y(x)=R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

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

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.

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<< Formula:KLS:14.07:09

formula in Dual q-Hahn

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