Formula:KLS:14.04:11

<< Formula:KLS:14.04:08

formula in Continuous q-Hahn

Formula:KLS:14.04:12 >>

${\displaystyle{\displaystyle{\displaystyle x{\widehat{p}}_{n}\!\left(x\right)={\widehat{p}}_{n+1}\!\left(x\right)+\frac{1}{2}\left[a{\mathrm{e}^{\mathrm{i}\phi}}+a^{-1}{\mathrm{e}^{-\mathrm{i}\phi}}-(A_{n}+C_{n})\right]{\widehat{p}}_{n}\!\left(x\right){}+\frac{1}{4}A_{n-1}C_{n}{\widehat{p}}_{n-1}\!\left(x\right)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{a{\mathrm{e}^{\mathrm{i}\phi}}(1-q^{n})(1-bcq^{n-1})(1-bdq^{n-1})(1-cd{\mathrm{e}^{-2\mathrm{i}\phi}}q^{n-1})}{(1-abcdq^{2n-2})(1-abcdq^{2n-1})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-ab{\mathrm{e}^{2\mathrm{i}\phi}}q^{n})(1-acq^{n})(1-adq^{n})(1-abcdq^{n-1})}{a{\mathrm{e}^{\mathrm{i}\phi}}(1-abcdq^{2n-1})(1-abcdq^{2n})}}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle{\widehat{p}}_{n}}}}$ : monic continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation in Section 14.4 of KLS.

URL links

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<< Formula:KLS:14.04:08

formula in Continuous q-Hahn

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