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≥1odd (-1)(^λ-1)/2q^λ/(1 - q^2λ) = 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 (^λ/3)q^λ/(1 - q^2λ) = 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. © 2002 Elsevier Science (USA).

Duke Scholars

Author Jayce Robert Getz Mathematics