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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Modular forms and $ k$-colored generalized Frobenius partitions


Authors: Heng Huat Chan, Liuquan Wang and Yifan Yang
Journal: Trans. Amer. Math. Soc. 371 (2019), 2159-2205
MSC (2010): Primary 05A17, 11F11; Secondary 11P83, 11F03, 11F33
DOI: https://doi.org/10.1090/tran/7447
Published electronically: September 20, 2018
MathSciNet review: 3894049
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Abstract: Let $ k$ and $ n$ be positive integers. Let $ c\phi _{k}(n)$ denote the number of $ k$-colored generalized Frobenius partitions of $ n$ and let $ \mathrm {C}\Phi _k(q)$ be the generating function of $ c\phi _{k}(n)$. In this article, we study $ \mathrm {C}\Phi _k(q)$ using the theory of modular forms and discover new surprising properties of $ \mathrm {C}\Phi _k(q)$.


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Additional Information

Heng Huat Chan
Affiliation: Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076, Singapore
Address at time of publication: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
Email: matchh@nus.edu.sg

Liuquan Wang
Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, People’s Republic of China
Email: mathlqwang@163.com; wangliuquan@u.nus.edu; wanglq@whu.edu.cn

Yifan Yang
Affiliation: Department of Mathematics, National Taiwan University, Taipei, Taiwan 10617
Email: yangyifan@ntu.edu.tw

DOI: https://doi.org/10.1090/tran/7447
Keywords: Generalized Frobenius partitions, generating functions, congruences, theta functions
Received by editor(s): June 9, 2017
Received by editor(s) in revised form: October 12, 2017
Published electronically: September 20, 2018
Additional Notes: Liuquan Wang is the corresponding author
This work was completed during the first author’s stay at the Faculty of Mathematics, University of Vienna. The first author would like to thank his host, Professor C. Krattenthaler, for his hospitality and for providing an excellent research environment during his stay in Vienna.
The second author was supported by the National Natural Science Foundation of China (11801424), the Fundamental Research Funds for the Central Universities (Grant No. 1301–413000053) and a start-up research grant (No. 1301–413100048) of the Wuhan University.
The third author was partially supported by Grant 102-2115-M-009-001-MY4 of the Ministry of Science and Technology, Taiwan (R.O.C.)
Dedicated: Dedicated to Professor George E. Andrews on the occasion of his 80th birthday
Article copyright: © Copyright 2018 American Mathematical Society