Formula:6684

Digital Library of Mathematical Functions ID 18.28.E13

${\displaystyle{\displaystyle C_{n}\left(\cos\theta;\beta\,|\,q\right)=\sum_{\ell=0}^{n}\frac{\left(\beta;q\right)_{\ell}\left(\beta;q\right)_{n-\ell}}{\left(q;q\right)_{\ell}\left(q;q\right)_{n-\ell}}e^{\mathrm{i}(n-2\ell)\theta}=\frac{\left(\beta;q\right)_{n}}{\left(q;q\right)_{n}}e^{\mathrm{i}n\theta}{{}_{2}\phi_{1}}\left({q^{-n},\beta\atop\beta^{-1}q^{1-n}};q,\beta^{-1}qe^{-2\mathrm{i}\theta}\right).}}$

Symbols List

• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}$:
• ${\displaystyle{\displaystyle{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)}}$: basic hypergeometric (or $$q$$ -hypergeometric) function
• ${\displaystyle{\displaystyle q}}$: real variable
• ${\displaystyle{\displaystyle\ell}}$: nonnegative integer
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
 

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