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Poincaré \(\boldsymbol{\alpha }\)-Series for Classical Schottky Groups

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Analytic Number Theory, Approximation Theory, and Special Functions

Abstract

The Poincaré α-series (\(\alpha \in {\mathbb{R}}^{n}\)) for classical Schottky groups are introduced and used to solve Riemann–Hilbert problems for n-connected circular domains. The classical Poincaré θ 2-series is a partial case of the α-series when α vanishes. The real Jacobi inversion problem and its generalizations are investigated via the Poincaré α-series. In particular, it is shown that the Riemann theta function coincides with the Poincaré α-series. Relations to conformal mappings of the multiply connected circular domains onto slit domains and the Schottky–Klein prime function are established. A fast algorithm to compute Poincaré series for disks close to each other is outlined.

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Acknowledgements

The author is grateful to D. Crowdy and T. DeLillo for helpful discussions, E.A. Krushevski for discussions concerning the results [41], and A.E. Malevich for the help in preparation of the code (see Appendix).

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Authors and Affiliations

  1. Department of Computer Sciences and Computer Methods, Pedagogical University, ul. Podchorazych 2, Krakow, 30-084, Poland

    Vladimir V. Mityushev

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  1. Vladimir V. Mityushev
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Correspondence to Vladimir V. Mityushev .

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  1. Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia

    Gradimir V. Milovanović

  2. Department of Mathematics, ETH Zürich, Zürich, Switzerland

    Michael Th. Rassias

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Dedicated to Professor Hari M. Srivastava

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Mityushev, V.V. (2014). Poincaré \(\boldsymbol{\alpha }\)-Series for Classical Schottky Groups. In: Milovanović, G., Rassias, M. (eds) Analytic Number Theory, Approximation Theory, and Special Functions. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0258-3_33

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