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Comptes Rendus Mathématique
Volume 353, n° 12
pages 1067-1073 (décembre 2015)
Doi : 10.1016/j.crma.2015.09.028
Received : 2 April 2015 ;  accepted : 29 September 2015
On the mock-theta behavior of Appell–Lerch series
Sur le comportement mock thêta de séries d'Appell–Lerch
 

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

Abstract

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.

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Résumé

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.

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