Formula:KLS:14.04:21

<< Formula:KLS:14.04:20

formula in Continuous q-Hahn

Formula:KLS:14.04:22 >>

${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{a{\mathrm{e}^{\mathrm{i}(\theta+2\phi)}},b{\mathrm{e}^{\mathrm{i}(\theta+2\phi)}}}{ab{\mathrm{e}^{2\mathrm{i}\phi}}}{q}{{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}}t}{}\qHyperrphis{2}{1}@@{c{\mathrm{e}^{-\mathrm{i}(\theta+2\phi)}},d{\mathrm{e}^{-\mathrm{i}(\theta+2\phi)}}}{cd{\mathrm{e}^{-2\mathrm{i}\phi}}}{q}{{\mathrm{e}^{\mathrm{i}(\theta+\phi)}}t}{}=\sum_{n=0}^{\infty}\frac{p_{n}\!\left(x;a,b,c,d;q\right)t^{n}}{\left(ab{\mathrm{e}^{2\mathrm{i}\phi}},cd{\mathrm{e}^{-2\mathrm{i}\phi}},q;q\right)_{n}}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqHahn
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.4 of KLS.

URL links

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formula in Continuous q-Hahn

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