Convergence of simultaneous Hermite-Padé approximants to the \(n\)-tuple of \(q\)-hypergeometric series \(\{_ 1\Phi_ 1 (_{c,\gamma_ j}^{(1,1)};z)\}_{j=1}^ n\) (Q688114)

Simultaneous Hermite-Padé approximation of the \(n\)-tuple \((f_ 1,\dots,f_ n)\) of power series \[ f_ j(z)= \sum_{k=0}^ \infty C_ k^{(j)} z^ k \qquad (j=1,\dots,n), \] where \(C_ k^{(j)}= \prod_{p=0}^{k-1} (C-q^{r_ j+p} )^{-1}\), \(\prod_ \emptyset :=1\), is investigated. Under certain conditions on \(C\), \(q\) and \(r_ j\) the \(f_ j\) have a common circle of convergence being the natural boundary. It is shown that ``close-to-diagonal'' and some other sequences of Hermite-Padé approximants converge in capacity to \((f_ 1,\dots, f_ n)\) inside this circle. The result is obtained from information about the zero distribution of the common denominator polynomial in the \(n\)-tuple of approximants.