Formula:KLS:14.29:04

<< Formula:KLS:14.29:03

formula in Discrete q-Hermite II

Formula:KLS:14.29:06 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{k=-\infty}^{\infty}\left[\tilde{h}_{m}\!\left(cq^{k};q\right)\tilde{h}_{n}\!\left(cq^{k};q\right)+\tilde{h}_{m}\!\left(-cq^{k};q\right)\tilde{h}_{n}\!\left(-cq^{k};q\right)\right]w(cq^{k};q)q^{k}{}=2\frac{\left(q^{2},-c^{2}q,-c^{-2}q;q^{2}\right)_{\infty}}{\left(q,-c^{2},-c^{-2}q^{2};q^{2}\right)_{\infty}}\frac{\left(q;q\right)_{n}}{q^{n^{2}}}\,\delta_{m,n}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle c>0}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle w(x;q)=\frac{1}{\left(\mathrm{i}x,-\mathrm{i}x;q\right)_{\infty}}=\frac{1}{\left(-x^{2};q^{2}\right)_{\infty}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation in Section 14.29 of KLS.

URL links

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<< Formula:KLS:14.29:03

formula in Discrete q-Hermite II

Formula:KLS:14.29:06 >>