ON THETA-TYPE FUNCTIONS IN THE FORM (x;q)∞

doi:10.1007/s10473-021-0617-z

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (6): 2086-2106.doi: 10.1007/s10473-021-0617-z

ON THETA-TYPE FUNCTIONS IN THE FORM (x;q)_∞

Changgui ZHANG

Laboratoire P. Painlevé(UMR-CNRS 8524), Département de mathématiques, FST, Université de Lille, Cité scientifique, 59655 Villeneuve d'Ascq cedex, France

收稿日期:2021-05-06 修回日期:2021-08-11 出版日期:2021-12-25 发布日期:2021-12-27
作者简介:Changgui ZHANG,E-mail:changgui.zhang@univ-lille.fr
基金资助:
The author was supported by Labex CEMPI (Centre Européen pour les Mathémmatiques, la Physique et leurs Interaction).

ON THETA-TYPE FUNCTIONS IN THE FORM (x;q)_∞

Changgui ZHANG

Laboratoire P. Painlevé(UMR-CNRS 8524), Département de mathématiques, FST, Université de Lille, Cité scientifique, 59655 Villeneuve d'Ascq cedex, France

Received:2021-05-06 Revised:2021-08-11 Online:2021-12-25 Published:2021-12-27
Supported by:
The author was supported by Labex CEMPI (Centre Européen pour les Mathémmatiques, la Physique et leurs Interaction).

摘要/Abstract

摘要： As in our previous work[14], a function is said to be of theta-type when its asymptotic behavior near any root of unity is similar to what happened for Jacobi theta functions. It is shown that only four Euler infinite products have this property. That this is the case is obtained by investigating the analyticity obstacle of a Laplace-type integral of the exponential generating function of Bernoulli numbers.

关键词: q-series, Mock theta-functions, Stokes phenomenon, continued fractions

Abstract: As in our previous work[14], a function is said to be of theta-type when its asymptotic behavior near any root of unity is similar to what happened for Jacobi theta functions. It is shown that only four Euler infinite products have this property. That this is the case is obtained by investigating the analyticity obstacle of a Laplace-type integral of the exponential generating function of Bernoulli numbers.

Key words: q-series, Mock theta-functions, Stokes phenomenon, continued fractions

中图分类号:

34M30

Changgui ZHANG. ON THETA-TYPE FUNCTIONS IN THE FORM (x;q)_∞[J]. 数学物理学报(英文版), 2021, 41(6): 2086-2106.

Changgui ZHANG. ON THETA-TYPE FUNCTIONS IN THE FORM (x;q)_∞[J]. Acta mathematica scientia,Series B, 2021, 41(6): 2086-2106.

参考文献

[1] Andrews G E, Richard R Askey, Ranjan R. Special functions//Encyclopedia of Mathematics and its Applications, 71. Cambridge:Cambridge University Press, 1999
[2] Berndt B C. Ramanujan's Notebooks. Part III. New York:Springer-Verlag, 1985
[3] Berndt B C. Ramanujan's Notebooks. Part IV. New York:Springer-Verlag, 1994
[4] Katsurada M. Asymptotic expansions of certain q-series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arithmetica, 2003, 107(3):269-298
[5] Fruchard A, Zhang C. Remarques sur les développements asymptotiques. Annales de la Faculté des sciences de Toulouse, Sér 6, 1999, 8:91-115
[6] Malgrange B. Sommation des séries divergentes. Exposition Math, 1995, 13(2/3):163-222
[7] Rademacher H. Topics in Analytic Number Theory. Springer-Verlag, 1973
[8] Ramis J -P. Gevrey asymptotics and applications to holomorphic ordinary differential equations//Differential Equations & Asymptotic Theory in Mathematical Physics (Wuhan Univ, China, 2003). Series in Analysis, 2. World Scientific, 2004:44-99
[9] Ramis J -P, Sauloy J, Zhang C. Local analytic classification of q-difference equations. Astérique, 2013, 355:vi+151 pages
[10] Watson G N. The final problem:An account of the mock theta functions. J London Math Soc, 1936, 11:55-80
[11] Whittaker E T, Watson G N. A Course of Modern Analysis. Fourth ed. Cambridge Univ Press, 1927
[12] Zagier D. Ramanujan's mock theta functions and their applications (after Zwegers and Ono-Bringmann), Séminaire Bourbaki. Vol 2007/2008. Astérisque, 2009, 326:143-164
[13] Zhang C. On the modular behaviour of the infinite product (1-x)(1-xq)(1-xq²)(1-xq³) …. C R Acad Sci Paris, Ser I, 2011, 349:725-730
[14] Zhang C. On the mock theta behavior of Appell-Lerch series. C R Acad Sci Paris, Ser I, 2015, 353(12):1067-1073
[15] Zhang C. A modular-type formula for (x; q)_∞. Ramanujan J, 2018, 46(1):269-305

ON THETA-TYPE FUNCTIONS IN THE FORM (x;q)_∞

ON THETA-TYPE FUNCTIONS IN THE FORM (x;q)_∞

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 3

Metrics

本文评价

推荐阅读 0

[1]	Lulu FANG, Jihua MA, Kunkun SONG, Xin YANG. MULTIFRACTAL ANALYSIS OF CONVERGENCE EXPONENTS FOR PRODUCTS OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS[J]. 数学物理学报(英文版), 2024, 44(4): 1594-1608.
[2]	房路路, 马际华, 宋昆昆, 吴敏. MULTIFRACTAL ANALYSIS OF THE CONVERGENCE EXPONENT IN CONTINUED FRACTIONS[J]. 数学物理学报(英文版), 2021, 41(6): 1896-1910.
[3]	胡动刚, 胡学海. ARBITRARILY LONG ARITHMETIC PROGRESSIONS FOR CONTINUED FRACTIONS OF LAURENT SERIES[J]. 数学物理学报(英文版), 2013, 33(4): 943-949.