∫ 0 ∞ x α ( - x ; q ) ∞ ⁢ L m ( α ) ⁡ ( x ; q ) ⁢ L n ( α ) ⁡ ( x ; q ) ⁢ 𝑑 x = ( q - α ; q ) ∞ ( q ; q ) ∞ ⁢ ( q α + 1 ; q ) n ( q ; q ) n ⁢ q n ⁢ Γ ⁡ ( - α ) ⁢ Γ ⁡ ( α + 1 ) ⁢ δ m , n superscript subscript 0 superscript x α q-Pochhammer x q q-Laguerre-polynomial-L α m x q q-Laguerre-polynomial-L α n x q differential-d x q-Pochhammer superscript q α q q-Pochhammer q q q-Pochhammer superscript q α 1 q n q-Pochhammer q q n superscript q n Euler-Gamma α Euler-Gamma α 1 Kronecker m n {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{x^{\alpha}}{\left(-x;q% \right)_{\infty}}\mathop{L^{(\alpha)}_{m}\/}\nolimits\!\left(x;q\right)\mathop% {L^{(\alpha)}_{n}\/}\nolimits\!\left(x;q\right)\,dx{}=\frac{\left(q^{-\alpha};% q\right)_{\infty}}{\left(q;q\right)_{\infty}}\frac{\left(q^{\alpha+1};q\right)% _{n}}{\left(q;q\right)_{n}q^{n}}\mathop{\Gamma\/}\nolimits\!\left(-\alpha% \right)\mathop{\Gamma\/}\nolimits\!\left(\alpha+1\right)\,\delta_{m,n}}}