×

On certain results on generalized basic hypergeometric functions. (English) Zbl 1173.33309

The authors investigate the following generalization of the basic hypergeometric function: \[ {}_2\phi_1^\tau(a,b;c;\tau;q,z) =\frac{\Gamma_q(c)}{\Gamma_q(b)}\sum_{n=0}^\infty \frac{(a;q)_n\Gamma_q(b+\tau n)}{(q;q)_n\Gamma_q(c+\tau n)}z^n, \qquad |q|<1, \quad \tau\in\mathbb R_{>0}. \] They prove a \(q\)-integral representation of \({}_2\phi_1^\tau\), derive difference equations and contiguous relations for the function.

MSC:

33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
33C05 Classical hypergeometric functions, \({}_2F_1\)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
PDFBibTeX XMLCite