Formula:KLS:14.04:01

<< Continuous dual q-Hahn

formula in Continuous q-Hahn

Formula:KLS:14.04:02 >>

${\displaystyle{\displaystyle{\displaystyle\frac{(a{\mathrm{e}^{\mathrm{i}\phi}})^{n}p_{n}\!\left(x;a,b,c,d;q\right)}{\left(ab{\mathrm{e}^{2\mathrm{i}\phi}},ac,ad;q\right)_{n}}{}=\qHyperrphis{4}{3}@@{q^{-n},abcdq^{n-1},a{\mathrm{e}^{\mathrm{i}(\theta+2\phi)}},a{\mathrm{e}^{-\mathrm{i}\theta}}}{ab{\mathrm{e}^{2\mathrm{i}\phi}},ac,ad}{q}{q}}}}$

Substitution(s)

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

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.4 of KLS.

URL links

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<< Continuous dual q-Hahn

formula in Continuous q-Hahn

Formula:KLS:14.04:02 >>