Partition Identities and a Theorem of Zagier

In the study of partition theory and q-series, identities that relate series to infinite products are of great interest (such as the famous Rogers–Ramanujan identities). Using a recent result of Zagier, we obtain an infinite family of such identities that is indexed by the positive integers. For example, if m=1, then we obtain the classical Eisenstein series identity ∑λ⩾1 odd(−1)(λ−1)/2qλ(1−q2λ)=q∏n=1∞(1−q8n)4(1−q4n)2 . If m=2 and (·3) denotes the usual Legendre symbol modulo 3, then we obtain ∑λ⩾1(λ3)qλ(1−q2λ)=q∏n=1∞(1−qn)(1−q6n)6(1−q2n)2(1−q3n)3 . We describe some of the partition theoretic consequences of these identities. In particular, we find simple formulas that solve the well-known problem of counting the number of representations of an integer as a sum of an arbitrary number of triangular numbers.