Formula:KLS:14.03:09

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formula in Continuous dual q-Hahn

Formula:KLS:14.03:10 >>

${\displaystyle{\displaystyle{\displaystyle 2x{\tilde{p}}_{n}\!\left(x\right)=A_{n}{\tilde{p}}_{n+1}\!\left(x\right)+\left[a+a^{-1}-\left(A_{n}+C_{n}\right)\right]{\tilde{p}}_{n}\!\left(x\right)+C_{n}{\tilde{p}}_{n-1}\!\left(x\right)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle C_{n}=a(1-q^{n})(1-bcq^{n-1})}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=a^{-1}(1-abq^{n})(1-acq^{n})}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle{\tilde{p}}_{n}}}}$ : normalized continuous dual ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hahn polynomial ${\displaystyle{\displaystyle{\displaystyle{\tilde{p}}}}}$ : http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde

Bibliography

Equation in Section 14.3 of KLS.

URL links

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formula in Continuous dual q-Hahn

Formula:KLS:14.03:10 >>