ISSN:
1432-0940
Keywords:
Padé approximant
;
Theta function
;
Rogers-Szegö polynomial
;
Convergence regions
;
Distribution of zeros
;
poles
;
Primary
;
41A21
;
33A65
;
Secondary
;
30E05
;
30E10
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 $$G_n (z): = \sum\limits_{j = 0}^n {\left[ {\begin{array}{*{20}c} n \\ j \\ \end{array} } \right]} z^j , n = 0,1,2,....$$
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01890574
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