Formula:KLS:14.24:02

<< Formula:KLS:14.24:01

formula in Al-Salam-Carlitz I

Formula:KLS:14.24:03 >>

${\displaystyle{\displaystyle{\displaystyle\int_{a}^{1}\left(qx,a^{-1}qx;q\right)_{\infty}U^{(a)}_{m}\!\left(x;q\right)U^{(a)}_{n}\!\left(x;q\right)\,d_{q}x{}=(-a)^{n}(1-q)\left(q;q\right)_{n}\left(q,a,a^{-1}q;q\right)_{\infty}q^{\binomial{n}{2}}\,\delta_{m,n}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle a<0}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle U^{(n)}_{\alpha}}}}$ : Al-Salam-Carlitz I polynomial : http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.24 of KLS.

URL links

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<< Formula:KLS:14.24:01

formula in Al-Salam-Carlitz I

Formula:KLS:14.24:03 >>