Formula:KLS:14.08:27

<< Formula:KLS:14.08:26

formula in Al-Salam-Chihara

Formula:KLS:14.08:28 >>

${\displaystyle{\displaystyle{\displaystyle\left(t;q\right)_{\infty}\cdot\qHyperrphis{2}{1}@@{a{\mathrm{e}^{\mathrm{i}\theta}},a{\mathrm{e}^{-\mathrm{i}\theta}}}{ab}{q}{t}{}=\sum_{n=0}^{\infty}\frac{(-1)^{n}a^{n}q^{\binomial{n}{2}}}{\left(ab,q;q\right)_{n}}Q_{n}\!\left(x;a,b\,|\,q\right)t^{n}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-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
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle Q_{n}}}}$ : Al-Salam-Chihara polynomial : http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.8 of KLS.

URL links

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<< Formula:KLS:14.08:26

formula in Al-Salam-Chihara

Formula:KLS:14.08:28 >>