Formula:6667

Digital Library of Mathematical Functions ID 18.27.E20

${\displaystyle{\displaystyle\int_{0}^{\infty}S_{n}\left(q^{\frac{1}{2}}x;q\right)S_{m}\left(q^{\frac{1}{2}}x;q\right)\exp\left(-\frac{(\ln x)^{2}}{2\ln\left(q^{-1}\right)}\right)\mathrm{d}x=\frac{\sqrt{2\pi q^{-1}\ln\left(q^{-1}\right)}}{q^{n}\left(q;q\right)_{n}}\delta_{n,m}.}}$

Symbols List

• ${\displaystyle{\displaystyle\delta_{\NVar{j},\NVar{k}}}}$:
• ${\displaystyle{\displaystyle S_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)}}$: Stieltjes–Wigert polynomial
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\exp\NVar{z}}}$: exponential function
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\ln\NVar{z}}}$: principal branch of logarithm function
• ${\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}$:
• ${\displaystyle{\displaystyle q}}$: real variable
• ${\displaystyle{\displaystyle m}}$: nonnegative integer
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
 

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