The Ramanujan Journal

, Volume 7, Issue 4, pp 407–434 | Cite as

An Identity of Ramanujan and the Representation of Integers as Sums of Triangular Numbers

  • Zhi-Guo Liu


Let k be a positive number and tk(n) denote the number of representations of n as a sum of k triangular numbers. In this paper, we will calculate t2k(n) in the spirit of Ramanujan. We first use the complex theory of elliptic functions to prove a theta function identity. Then from this identity we derive two Lambert series identities, one of them is a well-known identity of Ramanujan. Using a variant form of Ramanujan's identity, we study two classes of Lambert series and derive some theta function identities related to these Lambert series . We calculate t12(n), t16(n), t20(n), t24(n), and t28(n) using these Lambert series identities. We also re-derive a recent result of H. H. Chan and K. S. Chua [6] about t32(n). In addition, we derive some identities involving the Ramanujan function τ(n), the divisor function σ11(n), and t24(n). Our methods do not depend upon the theory of modular forms and are somewhat more transparent.

elliptic functions theta functions Lambert series triangular numbers Ramanujan τ(n)-function Jacobi's identity modular forms 


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© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Zhi-Guo Liu
    • 1
  1. 1.Department of MathematicsEast China Normal UniversityShanghaiPeople's Republic of China

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