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)/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 (λ/3)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. © 2002 Elsevier Science (USA).
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