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On Theta-Type Functions in the Form (x; q)∞

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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.

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References

  1. Andrews G E, Richard R Askey, Ranjan R. Special functions//Encyclopedia of Mathematics and its Applications, 71. Cambridge: Cambridge University Press, 1999

    Book  Google Scholar 

  2. Berndt B C. Ramanujan’s Notebooks. Part III. New York: Springer-Verlag, 1985

    Book  Google Scholar 

  3. Berndt B C. Ramanujan’s Notebooks. Part IV. New York: Springer-Verlag, 1994

    Book  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  5. Fruchard A, Zhang C. Remarques sur les développements asymptotiques. Annales de la Faculté des sciences de Toulouse, Sér 6, 1999, 2: 91–115

    Article  Google Scholar 

  6. Malgrange B. Sommation des séries divergentes. Exposition Math, 1995, 13(2/3): 163–222

    MathSciNet  MATH  Google Scholar 

  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, 2: 55–80

    Article  MathSciNet  Google Scholar 

  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, 2: 143–164

    MATH  Google Scholar 

  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, 2: 725–730

    Article  Google Scholar 

  14. Zhang C. On the mock theta behavior of Appell-Lerch series. C R Acad Sci Paris, Ser I, 2015, 353(12): 1067–1073

    Article  MathSciNet  Google Scholar 

  15. Zhang C. A modular-type formula for (x; q)∞. Ramanujan J, 2018, 46(1): 269–305

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

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

    Changgui Zhang

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  1. Changgui Zhang
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Correspondence to Changgui Zhang.

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Dedicated to the memory of Professor Jiarong YU

The author was supported by Labex CEMPI (Centre Europeen pour les Mathématiques, la Physique et leurs Interaction).

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Zhang, C. On Theta-Type Functions in the Form (x; q)∞. Acta Math Sci 41, 2086–2106 (2021). https://doi.org/10.1007/s10473-021-0617-z

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  • DOI: https://doi.org/10.1007/s10473-021-0617-z

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