Formula:KLS:14.07:29

<< Formula:KLS:14.07:28

formula in Dual q-Hahn

Formula:KLS:14.07:30 >>

${\displaystyle{\displaystyle{\displaystyle\lim_{\gamma\rightarrow 0}R_{n}\!\left(\mu(x);\gamma,c\gamma^{-1}q^{-N-1},N\right){q}=K_{n}\!\left(\lambda(x);c,N|q\right)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$ &

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

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

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
${\displaystyle{\displaystyle{\displaystyle K_{n}}}}$ : dual ${\displaystyle{\displaystyle{\displaystyle q}}}$-Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk

Bibliography

Equation in Section 14.7 of KLS.

URL links

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formula in Dual q-Hahn

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