Formula:6666

Digital Library of Mathematical Functions ID 18.27.E19

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

Symbols List

• ${\displaystyle{\displaystyle\delta_{\NVar{j},\NVar{k}}}}$:
• ${\displaystyle{\displaystyle S_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)}}$: Stieltjes–Wigert polynomial
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\ln\NVar{z}}}$: principal branch of logarithm function
• ${\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}$:
• ${\displaystyle{\displaystyle\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}}}$:
• ${\displaystyle{\displaystyle q}}$: real variable
• ${\displaystyle{\displaystyle m}}$: nonnegative integer
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
 

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