Padé approximants, continued fractions and Heine’s \(q\)-hypergeometric series. (English) Zbl 0784.33008

The author investigates the Padé approximants to Heine’s basic hypergometric function defined by \[ _ 2\Phi_ 1(a,b;c;x)=\sum^ \infty_ 0{[a]_ n[b]_ n\over[q]_ n[c]_ n}x^ n,| q|<1,\;| x|<1, \] where \([\alpha]_ n=(1-\alpha)(1-\alpha q)\ldots(1- \alpha q^{n-1})\), \(n\geq 1\) \([\alpha]_ 0=1\). Numerical results are provided which illustrate the superiority of the Padé approximations over Taylor series approximations.


41A21 Padé approximation