Formula:KLS:14.25:02

<< Formula:KLS:14.25:01

formula in Al-Salam-Carlitz II

Formula:KLS:14.25:03 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{q^{k^{2}}a^{k}}{\left(q;q\right)_{k}\left(aq;q\right)_{k}}V^{(a)}_{m}\!\left(q^{-k};q\right)V^{(a)}_{n}\!\left(q^{-k};q\right){}=\frac{\left(q;q\right)_{n}a^{n}}{\left(aq;q\right)_{\infty}q^{n^{2}}}\,\delta_{m,n}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle 0<aq<1}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle V^{(n)}_{\alpha}}}}$ : Al-Salam-Carlitz II polynomial : http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.25 of KLS.

URL links

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<< Formula:KLS:14.25:01

formula in Al-Salam-Carlitz II

Formula:KLS:14.25:03 >>