Formula:KLS:14.08:03

<< Formula:KLS:14.08:02

formula in Al-Salam-Chihara

Formula:KLS:14.08:04 >>

${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;a,b\,|\,q\right)=\left(b{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{n}{\mathrm{e}^{\mathrm{i}n\theta}}\,\qHyperrphis{2}{1}@@{q^{-n},a{\mathrm{e}^{\mathrm{i}\theta}}}{b^{-1}q^{-n+1}{\mathrm{e}^{\mathrm{i}\theta}}}{q}{b^{-1}q{\mathrm{e}^{-\mathrm{i}\theta}}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle Q_{n}}}}$ : Al-Salam-Chihara polynomial : http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1

Bibliography

Equation in Section 14.8 of KLS.

URL links

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<< Formula:KLS:14.08:02

formula in Al-Salam-Chihara

Formula:KLS:14.08:04 >>