Number theory/Ordinary differential equations
On the mock-theta behavior of Appell–Lerch series
Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1067-1073.

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:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.09.028
Zhang, Changgui 1

1 Laboratoire Paul-Painlevé, CNRS UMR 8524, UFR de mathématiques, Université Lille-1 (USTL), Cité scientifique, 59655 Villeneuve d'Ascq cedex, France
@article{CRMATH_2015__353_12_1067_0,
     author = {Zhang, Changgui},
     title = {On the mock-theta behavior of {Appell{\textendash}Lerch} series},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1067--1073},
     publisher = {Elsevier},
     volume = {353},
     number = {12},
     year = {2015},
     doi = {10.1016/j.crma.2015.09.028},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2015.09.028/}
}
TY  - JOUR
AU  - Zhang, Changgui
TI  - On the mock-theta behavior of Appell–Lerch series
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 1067
EP  - 1073
VL  - 353
IS  - 12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2015.09.028/
DO  - 10.1016/j.crma.2015.09.028
LA  - en
ID  - CRMATH_2015__353_12_1067_0
ER  - 
%0 Journal Article
%A Zhang, Changgui
%T On the mock-theta behavior of Appell–Lerch series
%J Comptes Rendus. Mathématique
%D 2015
%P 1067-1073
%V 353
%N 12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2015.09.028/
%R 10.1016/j.crma.2015.09.028
%G en
%F CRMATH_2015__353_12_1067_0
Zhang, Changgui. 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. http://www.numdam.org/articles/10.1016/j.crma.2015.09.028/

[1] Andrews, G.E. Mordell integrals and Ramanujan's “lost” notebook (Knopp, M.I., ed.), Analytic Number Theory, LNM, vol. 899, Springer-Verlag, 1981, pp. 10-48

[2] Andrews, G.E.; Hickerson, D. Ramanujan's “lost” notebook VII: the sixth-order mock theta functions, Adv. Math., Volume 89 (1991), pp. 60-105

[3] Balser, W. Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations, Universitext, Springer-Verlag, New York, 2000

[4] Choi, Y.-S. The basic bilateral hypergeometric series and the mock theta functions, Ramanujan J., Volume 24 (2011), pp. 345-386

[5] Di Vizio, L.; Zhang, C. On q-summation and confluence, Ann. Inst. Fourier, Volume 59 (2009) no. 1, pp. 347-392

[6] Duke, W. Almost a century of answering the question: what is a mock theta function?, Not. Amer. Math. Soc., Volume 61 (2014), pp. 1314-1320

[7] Gordon, B.; McIntosh, R.J. A survey of classical mock theta functions (Alladi, K.; Garvan, F., eds.), Partitions, q-Series, and Modular Forms, Dev. Math., vol. 23, 2011, pp. 95-144

[8] Hickerson, D.; Mortenson, E.T. Hecke-type double sums, Appell–Lerch sums, and mock theta functions, I, Proc. Lond. Math. Soc. (3), Volume 109 (2014), pp. 382-422

[9] Lerch, M. Bemerkungen zur Theorie der elliptischen Funktionen, Jahrb. Fortschr. Math., Volume 24 (1892), pp. 442-445

[10] Mordell, L.J. The definite integral eax2+bxecx+ddx and the analytic theory of numbers, Acta Math., Volume 61 (1933), pp. 323-360

[11] Ono, K. Personal reflections, and Gordon's work on modular forms and mock theta functions, Not. Amer. Math. Soc., Volume 60 (2013), pp. 862-863

[12] Ramis, J.-P. Gevrey asymptotics and applications to holomorphic ordinary differential equations, Wuhan Univ., China, 2003 (Series in Analysis), Volume vol. 2, World Scientific (2004), pp. 44-99

[13] Ramis, J.-P.; Sauloy, J.; Zhang, C. Local analytic classification of q-difference equations, Astérisque, Volume 355 (2013)

[14] Ramis, J.-P.; Zhang, C. Développements asymptotiques q-Gevrey et fonction thêta de Jacobi, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2002), pp. 277-280

[15] Watson, G.N. A theory of asymptotic series, Philos. Trans. R. Soc. (A), Volume 211 (1911), pp. 279-313

[16] Watson, G.N. The final problem: an account of the mock theta functions, J. Lond. Math. Soc., Volume 11 (1936), pp. 55-80

[17] Whittaker, E.T.; Watson, G.N. A Course of Modern Analysis, Cambridge University Press, 1927

[18] Zagier, D. Ramanujan's mock theta functions and their applications (after Zwegers and Ono-Bringmann), Astérisque, Volume 326 (2009), pp. 143-164

[19] Zhang, C. Développements asymptotiques q-Gevrey et séries Gq-sommables, Ann. Inst. Fourier, Volume 49 (1999), pp. 227-261

[20] Zhang, C. Une sommation discrète pour des équations aux q-différences linéaires et à coefficients analytiques : théorie générale et exemples (Braaksma, B.L.J. et al., eds.), Differential Equations and the Stokes Phenomenon, World Scientific, 2002, pp. 309-329

[21] Zhou, S.; Luo, Z.; Zhang, C. On summability of formal solutions to a Cauchy problem and generalization of Mordell's theorem, C. R. Acad. Sci. Paris, Ser. I, Volume 348 (2010), pp. 753-758

[22] Zwegers, S.P. Mock θ-functions and real analytic modular forms, Urbana, IL, 2000 (Contemp. Math.), Volume vol. 291, American Mathematical Society, Providence, RI, USA (2001), pp. 269-277

Cited by Sources: