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Fractional powers of the generating function for the partition function

Volume 187 / 2019

Heng Huat Chan, Liuquan Wang Acta Arithmetica 187 (2019), 59-80 MSC: Primary 05A17; Secondary 11P83. DOI: 10.4064/aa180206-10-4 Published online: 29 November 2018

Abstract

Let $p_{k}(n)$ be the coefficient of $q^n$ in the series expansion of $(q;q)_{\infty}^{k}$. It is known that the partition function $p(n)$, which corresponds to the case when $k=-1$, satisfies congruences such as $p(5n+4)\equiv 0\pmod{5}$. In this article, we discuss congruences satisfied by $p_{k}(n)$ when $k$ is a rational number.

Authors

  • Heng Huat ChanDepartment of Mathematics
    National University of Singapore
    Singapore, 119076, Singapore
    e-mail
  • Liuquan WangSchool of Mathematics and Statistics
    Wuhan University
    Wuhan 430072, Hubei, People’s Republic of China
    e-mail
    e-mail

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