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Convergence of simultaneous Hermite-Padé approximants to the $$n$$-tuple of $$q$$-hypergeometric series $$\{_ 1\Phi_ 1 (_{c,\gamma_ j}^{(1,1)};z)\}_{j=1}^ n$$. (English) Zbl 0786.33010
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.
Reviewer: J.Müller (Trier)

##### MSC:
 33D99 Basic hypergeometric functions 30E10 Approximation in the complex plane 41A21 Padé approximation
##### Keywords:
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