Formula:KLS:14.19:22

<< Formula:KLS:14.19:21

formula in Continuous q-Laguerre

Formula:KLS:14.19:23 >>

${\displaystyle{\displaystyle{\displaystyle\left(t;q\right)_{\infty}\cdot\qHyperrphis{2}{1}@@{q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{-\mathrm{i}\theta}}}{q^{\alpha+1}}{q}{t}{}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binomial{n}{2}}}{\left(q^{\alpha+1};q\right)_{n}}P^{(\alpha)}_{n}\!\left(x|q\right)t^{n}}}}$

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
${\displaystyle{\displaystyle{\displaystyle P^{(n)}_{\alpha}}}}$ : continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Laguerre polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.19 of KLS.

URL links

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<< Formula:KLS:14.19:21

formula in Continuous q-Laguerre

Formula:KLS:14.19:23 >>