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On asymptotic formulas for certain $ q$-series involving partial theta functions


Authors: Sihun Jo and Byungchan Kim
Journal: Proc. Amer. Math. Soc. 143 (2015), 3253-3263
MSC (2010): Primary 11P82, 11P81
Published electronically: April 20, 2015
MathSciNet review: 3348769
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Abstract: In this article, we investigate the integer sequence $ a_d (n)$, where for a positive integer $ d$, $ a_d (n)$ is defined by

$\displaystyle \prod _{n=1}^{\infty } \frac {1}{1-q^n} \sum _{n=0}^{\infty } (-1)^n q^{ ( d n^2 - ( d - 2) n ) / 2 } = \sum _{n=0}^{\infty } a_d (n) q^n . $

In particular, we will present a combinatorial model for $ a_d (n)$ via integer partitions and this will play a crucial role in obtaining an asymptotic formula for $ a_d (n)$. Moreover, we will show that $ p(n) - 2 a_d (n)$ has an unexpected sign pattern via combinatorial arguments and asymptotic formulas.

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Additional Information

Sihun Jo
Affiliation: Department of Mathematics, Yonsei University, Seoul 120-749, Republic of Korea
Address at time of publication: School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro Dongdaemun-Gu Seoul 130-722, Republic of Korea
Email: topspace@kias.re.kr

Byungchan Kim
Affiliation: School of Liberal Arts and Institute of Convergence Fundamental Studies, Seoul National University of Science and Technology, 232 Gongneung-Ro, Nowongu, Seoul 139–743, Republic of Korea
Email: bkim4@seoultech.ac.kr

DOI: https://doi.org/10.1090/proc/12166
Keywords: Partial theta function, integer partitions, asymptotic formula
Received by editor(s): September 28, 2012
Received by editor(s) in revised form: October 1, 2012, January 7, 2013, and September 7, 2013
Published electronically: April 20, 2015
Additional Notes: This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science.
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2015 American Mathematical Society