Formula:6690

Digital Library of Mathematical Functions ID 18.28.E19

${\displaystyle{\displaystyle R_{n}(x)=R_{n}\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)=\sum_{\ell=0}^{n}\frac{q^{\ell}\left(q^{-n},\alpha\beta q^{n+1};q\right)_{\ell}}{\left(\alpha q,\beta\delta q,\gamma q,q;q\right)_{\ell}}\*\prod_{j=0}^{\ell-1}(1-q^{j}x+\gamma\delta q^{2j+1})={{}_{4}\phi_{3}}\left({q^{-n},\alpha\beta q^{n+1},q^{-y},\gamma\delta q^{y+1}\atop\alpha q,\beta\delta q,\gamma q};q,q\right),}}$

Constraint(s)

${\displaystyle{\displaystyle n=0,1,\dots,N}}$

Symbols List

• ${\displaystyle{\displaystyle{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)}}$: basic hypergeometric (or $$q$$ -hypergeometric) function
• ${\displaystyle{\displaystyle\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}}}$:
• ${\displaystyle{\displaystyle y}}$: real variable
• ${\displaystyle{\displaystyle q}}$: real variable
• ${\displaystyle{\displaystyle\ell}}$: nonnegative integer
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle N}}$: positive integer
• ${\displaystyle{\displaystyle\delta}}$: arbitary small positive constant
• ${\displaystyle{\displaystyle x}}$: real variable
 

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