On the mock-theta behavior of Appell–Lerch series

Changgui Zhang

doi:10.1016/j.crma.2015.09.028

Number theory/Ordinary differential equations

On the mock-theta behavior of Appell–Lerch series

Presented by: Jean-Pierre Ramis

Changgui Zhang ¹

¹ Laboratoire Paul-Painlevé, CNRS UMR 8524, UFR de mathématiques, Université Lille-1 (USTL), Cité scientifique, 59655 Villeneuve d'Ascq cedex, France

Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1067-1073.

Abstract
Résumé

The goal of this paper is to find one natural way to write the first order Appell–Lerch series in terms of two functions whose asymptotic behavior becomes simple. It is shown that such writing exists, using only theta-like functions and functions having a Gevrey asymptotic expansion. In order of simplify the presentation, we introduce three types of theta-like functions that will be called theta-type, false theta-type and mock theta-type.

Le but de cette Note est de trouver une manière naturelle d'écrire chaque série d'Appell–Lerch du premier ordre au moyen de deux fonctions dont le comportement asymptotique devient plus simple. On démontre qu'une telle écriture existe, avec seulement des fonctions de type thêta et celles qui ont un développement asymptotique Gevrey. Afin de faciliter l'exposé, nous introduisons trois types de fonctions du genre thêta, qui seront appelés type thêta, type faux thêta et type mock thêta.

Received: 2015-04-02
Accepted: 2015-09-29
Published online: 2015-10-31

DOI: 10.1016/j.crma.2015.09.028

Author's affiliations:

Changgui Zhang ¹

¹ Laboratoire Paul-Painlevé, CNRS UMR 8524, UFR de mathématiques, Université Lille-1 (USTL), Cité scientifique, 59655 Villeneuve d'Ascq cedex, France

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     title = {On the mock-theta behavior of {Appell{\textendash}Lerch} series},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1067--1073},
     publisher = {Elsevier},
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     year = {2015},
     doi = {10.1016/j.crma.2015.09.028},
     language = {en},
}

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Changgui Zhang. On the mock-theta behavior of Appell–Lerch series. Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1067-1073. doi : 10.1016/j.crma.2015.09.028. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.09.028/

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