Formula:KLS:14.27:02

<< Formula:KLS:14.27:01

formula in Stieltjes-Wigert

Formula:KLS:14.27:03 >>

${\displaystyle{\displaystyle{\displaystyle\int_{0}^{\infty}\frac{S_{m}\!\left(x;q\right)S_{n}\!\left(x;q\right)}{\left(-x,-qx^{-1};q\right)_{\infty}}\,dx=-\frac{\ln q}{q^{n}}\frac{\left(q;q\right)_{\infty}}{\left(q;q\right)_{n}}\,\delta_{m,n}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle S_{n}}}}$ : Stieltjes-Wigert polynomial : http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.27 of KLS.

URL links

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<< Formula:KLS:14.27:01

formula in Stieltjes-Wigert

Formula:KLS:14.27:03 >>