Formula:KLS:14.26:20

<< Formula:KLS:14.26:19

formula in Continuous q-Hermite

Formula:KLS:14.26:21 >>

${\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}\frac{P^{(\alpha)}_{n}\!\left(x|q\right)}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}=\frac{H_{n}\!\left(x\,|\,q\right)}{\left(q;q\right)_{n}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle P^{(n)}_{\alpha}}}}$ : continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Laguerre polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre
${\displaystyle{\displaystyle{\displaystyle H_{n}}}}$ : continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hermite polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqHermite
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1

Bibliography

Equation in Section 14.26 of KLS.

URL links

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<< Formula:KLS:14.26:19

formula in Continuous q-Hermite

Formula:KLS:14.26:21 >>