Formula:KLS:14.21:36

<< Formula:KLS:14.21:35

formula in q-Laguerre

Formula:KLS:14.22:01 >>

${\displaystyle{\displaystyle{\displaystyle xL^{(1/2)}_{n}\!\left(x^{2};q^{2}\right)=\frac{(-1)^{n}q^{2n^{2}+n}}{\left(q^{2};q^{2}\right)_{n}}\tilde{h}_{2n+1}\!\left(x;q\right)}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle L_{n}^{(\alpha)}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Laguerre polynomial : http://drmf.wmflabs.org/wiki/Definition:qLaguerre
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}}}}$ : discrete ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hermite II polynomial : http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII

Bibliography

Equation in Section 14.21 of KLS.

URL links

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<< Formula:KLS:14.21:35

formula in q-Laguerre

Formula:KLS:14.22:01 >>