Infinite families of arithmetic identities for 4-cores

Authors

  • Nayandeep Deka Baruah Tezpur University
  • Kallol Nath Tezpur University

Keywords:

partitions, $t$-cores, theta functions, dissection

Abstract

Let $u(n)$ and $v(n)$ be the numbers of representations of a nonnegative integer $n$ in the forms $x^2+4y^2+4z^{2}$ and $x^2+2y^2+2z^{2}$, respectively, with $x,y,z\in\mathbb{Z}$, and let $a_4(n)$ and $r_3(n)$ be the number of $4$-cores of $n$ and the number of representations of $n$ as a sum of three squares, respectively. By employing simple theta function identities of Ramanujan, we prove that $u(8n+5)=8a_4(n)=v(8n+5)=\frac{1}{3}r_3(8n+5)$. With the help of this and a classical result of Gauss, we find a simple proof of a result on $a_4(n)$ proved earlier by Ono and Sze. We also find some new infinite families of arithmetic relations involving $a_4(n)$. 10.1017/S0004972712000378

Published

2013-02-24

Issue

Vol. 87 No. 2 (2013)

Section

Articles
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