INFINITE FAMILIES OF ARITHMETIC IDENTITIES FOR 4-CORES

Abstract

AbstractLetu(n) andv(n) be the number of representations of a nonnegative integernin the formsx2+4y2+4z2andx2+2y2+2z2, respectively, withx,y,z∈ℤ, and leta4(n) andr3(n) be the number of 4-cores ofnand the number of representations ofnas 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 ona4(n) proved earlier by K. Ono and L. Sze [‘4-core partitions and class numbers’,Acta Arith.80(1997), 249–272]. We also find some new infinite families of arithmetic relations involvinga4(n) .