Summary
We investigate the convergence of sequences of Padé approximants for the power series
where
For “most”A, and |C| ≠ 1, we show that, ifq = e iθ whereθ ∈[0, 2π) andθ/2π is irrational,f(z) has a natural boundary on its circle of convergence. We show that diagonal and other sequences of Padé approximants converge in capacity tof and further obtain subsequences of the diagonal sequences{[n/n](z)} ∞n=1 that converge locally uniformly.
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Driver, K.A., Lubinsky, D.S. Convergence of Padé approximants for aq-hypergeometric series (wynn's power series III). Aequat. Math. 45, 1–23 (1993). https://doi.org/10.1007/BF01844422
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DOI: https://doi.org/10.1007/BF01844422