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Infinite families of congruences modulo 5 and 9 for overpartitions

Volume 66 / 2018

Chao Gu, Michael D. Hirschhorn, James A. Sellers, Ernest X. W. Xia Bulletin Polish Acad. Sci. Math. 66 (2018), 31-44 MSC: 11P83, 05A17. DOI: 10.4064/ba8129-9-2017 Published online: 23 October 2017

Abstract

Let $\bar{p}(n)$ denote the number of overpartitions of $n$. Recently, a number of congruences modulo 5 and powers of 3 for $\bar{p}(n)$ were established by a number of authors. In particular, Treneer proved that the generating function for $\bar{p}(5n)$ modulo 5 is $\sum_{n=0}^\infty \bar{p}(5n)q^n \equiv {(q;q)_\infty^6 }/{(q^2;q^2)_\infty^3} \pmod 5.$ In this paper, employing elementary methods, we establish the generating function of $\bar{p}(5n)$ which yields the congruence due to Treneer. Furthermore, we prove some new congruences modulo 5 and 9 for $\bar{p}(n)$ by utilizing the fact that the generating functions for $\bar{p}(5n)$ modulo 5 and for $\bar{p}(3n)$ modulo 9 are eigenforms for half-integral weight Hecke operators.

Authors

  • Chao GuDepartment of Mathematics
    Jiangsu University
    Jiangsu, Zhenjiang 212013, P.R. China
    e-mail
  • Michael D. HirschhornSchool of Mathematics and Statistics
    University of New South Wales
    Sydney 2052, Australia
    e-mail
  • James A. SellersDepartment of Mathematics
    Penn State University
    University Park, PA 16802, U.S.A.
    e-mail
  • Ernest X. W. XiaDepartment of Mathematics
    Jiangsu University
    Jiangsu, Zhenjiang 212013, P. R. China
    e-mail

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