INFINITE FAMILIES OF ARITHMETIC IDENTITIES FOR 4-CORES

Let u(n) and v(n) be the number 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 is an element of 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) = 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 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 involving a(4)(n).