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Asymptotic expansions of certain partial theta functions


Authors: Bruce C. Berndt and Byungchan Kim
Journal: Proc. Amer. Math. Soc. 139 (2011), 3779-3788
MSC (2010): Primary 11F27, 33D15; Secondary 11B68
DOI: https://doi.org/10.1090/S0002-9939-2011-11062-1
Published electronically: July 7, 2011
MathSciNet review: 2823024
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Abstract: We establish an asymptotic expansion for a class of partial theta functions generalizing a result found in Ramanujan's second notebook. Properties of the coefficients in this more general asymptotic expansion are studied, with connections made to combinatorics and a certain Dirichlet series.


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Additional Information

Bruce C. Berndt
Affiliation: Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801
Email: berndt@illinois.edu

Byungchan Kim
Affiliation: School of Liberal Arts, Seoul National University of Science and Technology, 172 Gongreung 2 dong, Nowongu, Seoul, 139-743, Republic of Korea
Email: bkim4@seoultech.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-2011-11062-1
Keywords: Theta functions, partial theta functions, false theta functions, asymptotic expansion, Ramanujan’s notebooks, Euler numbers, Hermite polynomials, Dirichlet series associated with a polynomial
Received by editor(s): April 5, 2010
Published electronically: July 7, 2011
Additional Notes: The first author’s research was partially supported by grant No. H98230-07-1-0088 from the National Security Agency.
Part of this work was done while the second author was at the Korea Institute of Advanced Study
Communicated by: Jim Haglund
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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