Formula:KLS:14.29:06

<< Formula:KLS:14.29:04

formula in Discrete q-Hermite II

Formula:KLS:14.29:07 >>

${\displaystyle{\displaystyle{\displaystyle\int_{-\infty}^{\infty}\frac{\tilde{h}_{m}\!\left(x;q\right)\tilde{h}_{n}\!\left(x;q\right)}{\left(-x^{2};q^{2}\right)_{\infty}}\,d_{q}x=\frac{\left(q^{2},-q,-q;q^{2}\right)_{\infty}}{\left(q^{3},-q^{2},-q^{2};q^{2}\right)_{\infty}}\frac{\left(q;q\right)_{n}}{q^{n^{2}}}\,\delta_{m,n}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}}}}$ : discrete ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hermite II polynomial : http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.29 of KLS.

URL links

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<< Formula:KLS:14.29:04

formula in Discrete q-Hermite II

Formula:KLS:14.29:07 >>