Math @ Duke

Publications [#292894] of Jayce R. Getz
Papers Published
 Getz, J; Mahlburg, K, Partition identities and a theorem of Zagier,
Journal of Combinatorial Theory, Series A, vol. 100 no. 1
(2002),
pp. 2743, ISSN 00973165 [doi]
(last updated on 2018/07/22)
Abstract: In the study of partition theory and qseries, identities that relate series to infinite products are of great interest (such as the famous RogersRamanujan 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)(λ1)/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 wellknown 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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