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On the spectrum of a partial theta function

Published online by Cambridge University Press:  03 October 2014

Vladimir Petrov Kostov
Vladimir Petrov Kostov*
Affiliation:
Laboratoire de Mathématiques, Université de Nice, Parc Valrose, 06108 Nice, Cedex 2, France, ([email protected])
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Abstract

The bivariate series defines a partial theta function. For fixed q, θ(q, ·) is an entire function. We show that for ∣q∣ ≤ 0.108 the function θ(q, ·) has no multiple zeros.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 5 , October 2014 , pp. 925 - 933
Copyright
Copyright © Royal Society of Edinburgh 2014 

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References

1Andrews, G. E. and Berndt, B. C.. Ramanujan's lost notebook, part II (Springer, 2009).Google Scholar
2Berndt, B. C. and Kim, B.. Asymptotic expansions of certain partial theta functions. Proc. Am. Math. Soc. 139 (2011), 3779–3788.Google Scholar
3Bringmann, K., Folsom, A. and Rhoades, R. C.. Partial theta functions and mock modular forms as q-hypergeometric series. Ramanujan J. 29 (2012), 295–310.CrossRefGoogle Scholar
4Katkova, O. M., Lobova, T. and Vishnyakova, A. M.. On power series having sections with only real zeros. Computat. Meth. Funct. Theory 3 (2003), 425–441.Google Scholar
5Katsnelson, V.. On summation of the Taylor series of the function 1/(1 – z) by the theta summation method. Preprint, arXiv:1402.4629v1 (2014).Google Scholar
6Kostov, V. P.. On the zeros of a partial theta function Bull. Sci. Math. 137 (2013), 1018–1030.Google Scholar
7Kostov, V. P.. Asymptotics of the spectrum of partial theta function. Rev. Mat. Complut. (In the press.)Google Scholar
8Kostov, V. P. and Shapiro, B.. Hardy–Petrovitch–Hutchinson's problem and partial theta function. Duke Math. J. 162 (2013), 825–861.Google Scholar
9Prellberg, T.. The combinatorics of the leading root of the partial theta function. Preprint, arXiv 1210.0095 (2012).Google Scholar
10Sokal, A.. The leading root of the partial theta function. Adv. Math. 229 (2012), 2603–2621.Google Scholar
11Warnaar, S. O.. Partial theta functions. I. Beyond the lost notebook. Proc. Lond. Math. Soc. 87 (2003), 363–395.Google Scholar