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FOUR IDENTITIES FOR THIRD ORDER MOCK THETA FUNCTIONS

Part of: Basic hypergeometric functions Additive number theory; partitions

Published online by Cambridge University Press:  21 September 2018

GEORGE E. ANDREWS ,
BRUCE C. BERNDT ,
SONG HENG CHAN ,
SUN KIM  and
AMITA MALIK
GEORGE E. ANDREWS
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA email gea1@psu.edu
BRUCE C. BERNDT
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA email berndt@illinois.edu
SONG HENG CHAN
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Republic of Singapore email ChanSH@ntu.edu.sg
SUN KIM
Affiliation:
Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany email skim9@uni-koeln.de
AMITA MALIK
Affiliation:
Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854, USA email amita.malik@rutgers.edu
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Abstract

In 2005, using a famous lemma of Atkin and Swinnerton-Dyer (Some properties of partitions, Proc. Lond. Math. Soc. (3) 4 (1954), 84–106), Yesilyurt (Four identities related to third order mock theta functions in Ramanujan’s lost notebook, Adv. Math. 190 (2005), 278–299) proved four identities for third order mock theta functions found on pages 2 and 17 in Ramanujan’s lost notebook. The primary purpose of this paper is to offer new proofs in the spirit of what Ramanujan might have given in the hope that a better understanding of the identities might be gained. Third order mock theta functions are intimately connected with ranks of partitions. We prove new dissections for two rank generating functions, which are keys to our proof of the fourth, and the most difficult, of Ramanujan’s identities. In the last section of this paper, we establish new relations for ranks arising from our dissections of rank generating functions.

MSC classification

Primary: 33D15: Basic hypergeometric functions in one variable, $_rphi_s$ 11P83: Partitions; congruences and congruential restrictions
Type
Article
Information
Nagoya Mathematical Journal , Volume 239 , September 2020 , pp. 173 - 204
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Copyright
© 2018 Foundation Nagoya Mathematical Journal

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Footnotes

The third author was partially supported by the Singapore Ministry of Education Academic Research Fund, Tier 2, project number MOE2014-T2-1-051, ARC40/14

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