Formula:KLS:14.03:13

<< Formula:KLS:14.03:10

formula in Continuous dual q-Hahn

Formula:KLS:14.03:14 >>

${\displaystyle{\displaystyle{\displaystyle x{\widehat{p}}_{n}\!\left(x\right)={\widehat{p}}_{n+1}\!\left(x\right)+\frac{1}{2}\left[a+a^{-1}-(A_{n}+C_{n})\right]{\widehat{p}}_{n}\!\left(x\right){}+\frac{1}{4}(1-q^{n})(1-abq^{n-1}){}(1-acq^{n-1})(1-bcq^{n-1}){\widehat{p}}_{n-1}\!\left(x\right)}}}$

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
${\displaystyle{\displaystyle{\displaystyle{\widehat{p}}_{n}}}}$ : monic continuous dual ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn

Bibliography

Equation in Section 14.3 of KLS.

URL links

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<< Formula:KLS:14.03:10

formula in Continuous dual q-Hahn

Formula:KLS:14.03:14 >>