Convergence of Padé approximants for aq-hypergeometric series (Wynn's Power Series I)

Summary

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

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

where

$$a_j = \prod\limits_{k = 0}^{j - 1} { (A - q^{k + \alpha } )} , j \geqslant 1, A, \alpha cons\tan ts, q \varepsilon \mathbb{C}$$

. For “most”A, we show that ifq =e ^tθ whereθ ∈ [0, 2π) andθ/2π is irrational,f(z) has a natural boundary on its circle of convergence. We show that diagonal sequences of Padé approximants converge in capacity tof and further obtain subsequences of the diagonal sequence{[n/n](z)} ^∞ _{n = 1} which converge locally uniformly.