Formula:KLS:14.21:31

<< Formula:KLS:14.21:30

formula in q-Laguerre

Formula:KLS:14.21:34 >>

${\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}}=\frac{\left(q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}q^{n}},\qquad h_{0}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}}=-\frac{\left(q^{-\alpha};q\right)_{\infty}}{\left(q;q\right)_{\infty}}\frac{\pi}{\sin\left(\pi\alpha\right)}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h_{n}=q^{-\frac{1}{2}\alpha(\alpha+1)}\left(q;q\right)_{\alpha}\log(q^{-1})\qquad(\alpha\in\ZZ_{\geq 0})}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : sine function : http://dlmf.nist.gov/4.14#E1
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\mathrm{log}}}}$ : principle branch of logarithm logarithm : http://dlmf.nist.gov/4.2#E2

Bibliography

Equation in Section 14.21 of KLS.

URL links

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<< Formula:KLS:14.21:30

formula in q-Laguerre

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