Formula:6686

Digital Library of Mathematical Functions ID 18.28.E15

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "TeX input is invalid."): {\displaystyle \frac{1}{2\pi}\int_{0}^{\pi}\contqultrasphpoly{n}@{\cos@@{\theta}}{\beta}{q}\contqultrasphpoly{m}@{\cos@@{\theta}}{\beta}{q}\*\abs{\frac{\qPochhammer{e^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{\beta e^{2\iunit\theta}}{q}{\infty}}}^{2}\diff{\theta}=\frac{\qmultiPochhammersym{\beta,\beta q}{q}{\infty}}{\qmultiPochhammersym{\beta^{2},q}{q}{\infty}}\frac{(1-\beta)\qPochhammer{\beta^{2}}{q}{n}}{(1-\beta q^{n})\qPochhammer{q}{q}{n}}\Kroneckerdelta{n}{m},}

Constraint(s)

${\displaystyle{\displaystyle-1<\beta<1}}$

Symbols List

• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "TeX input is invalid."): {\displaystyle \Kroneckerdelta{\NVar{j}}{\NVar{k}}} :
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}$:
• ${\displaystyle{\displaystyle\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}}}$:
• ${\displaystyle{\displaystyle C_{\NVar{n}}\left(\NVar{x};\NVar{\beta}\,|\,\NVar{q}\right)}}$: continuous $$q$$ -ultraspherical polynomial
• ${\displaystyle{\displaystyle q}}$: real variable
• ${\displaystyle{\displaystyle m}}$: nonnegative integer
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
 

Resources that cite this formula

No records found.