Convergence of Padé approximants for aq-hypergeometric series (wynn's power series III)

Summary

We investigate the convergence of sequences of Padé approximants for the power series

$$f(z) = 1 + \sum\limits_{j = 1}^\infty {a_j z^j } $$

where

$$a_j = \prod\limits_{k = 0}^{j - 1} {\frac{{(A - q^{k + \alpha } )}}{{(C - q^{k + \gamma } )}},} j \geqslant 1;\alpha ,\gamma \in \mathbb{R};A,C,q \in \mathbb{C}.$$

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.