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Convergence of Padé approximants of partial theta functions and the Rogers-Szegö polynomials

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Abstract

We investigate the convergence of sequences of Padé approximants for the partial theta function

$$h_q (z): = \sum\limits_{j = 0}^\infty { q^{j(j - 1)/2_{Z^j } } } , q = e^{i\theta } , \theta \in [0,2\pi ).$$

Whenθ/(2π) is irrational, this function has the unit circle as its natural boundary. We determine subrogions of ¦z¦ < 1 in which sequences of Padé approximants converge uniformly, and subrogions in which they converge in capacity, but not uniformly. In particular, we show that only a proper subsequence of the diagonal sequence {[n/n]} ∞ n=1 converges locally uniformly in all of ¦z¦< l; in contrast, no subsequence of any Padé row {[m/n]} ∞ m=1 (withn ≥ 2 fixed) can converge locally uniformly in all of ¦z¦ < 1. Further, we obtain the zero and pole distributions of sequences of Padé approximants by analyzing the zero distribution of the Rogers-Szegö polynomials

$$G_n (z): = \sum\limits_{j = 0}^n {\left[ {\begin{array}{*{20}c} n \\ j \\ \end{array} } \right]} z^j , n = 0,1,2,....$$

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Authors and Affiliations

  1. National Research Institute for Mathematical Sciences, C.S.I.R., P.O. Box 395, 0001, Pretoria, Republic of South Africa

    D. S. Lubinsky

  2. Institute for Constructive Mathematics Department of Mathematics, University of South Florida, 33620, Tampa, Florida, USA

    E. B. Saff

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  1. D. S. Lubinsky
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  2. E. B. Saff
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Communicated by Paul Nevai.

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Lubinsky, D.S., Saff, E.B. Convergence of Padé approximants of partial theta functions and the Rogers-Szegö polynomials. Constr. Approx 3, 331–361 (1987). https://doi.org/10.1007/BF01890574

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  • DOI: https://doi.org/10.1007/BF01890574

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