Proofs of some conjectures of Sun on the relations between sums of squares and sums of triangular numbers

摘要

In a recent paper, Sun posed six conjectures on the relations between T(a(1), a(2), ..., a(k); n) and N(a(1), a(2), ..., a(k); n), where T(a(1), a(2), ..., a(k); n) denotes the number of representations of n as a(1) x(1)(x(1)+1)/2 +a(2) x(2)(x(2)+1)/2+...+a(k) x(k)(x(k)+1/2 where a(1), a(2), ..., a(k) axe positive integers, n, x(1), x(2), ..., x(k) are arbitrary nonnegative integers, and N(a(1), a(2), ..., a(k); n) denotes the number of representations of n as a(1)x(1)(2) + a(2)x(2)(2) +...+ a(k)x(k)(2), where this time x(1), x(2),... , x(k) are integers. In this paper, we prove Sun's six conjectures by using Ramanujan's theta function identities.