Abstract
Let k be a positive number and t k(n) denote the number of representations of n as a sum of k triangular numbers. In this paper, we will calculate t 2k (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 t 12(n), t 16(n), t 20(n), t 24(n), and t 28(n) using these Lambert series identities. We also re-derive a recent result of H. H. Chan and K. S. Chua [6] about t 32(n). In addition, we derive some identities involving the Ramanujan function τ(n), the divisor function σ11(n), and t 24(n). Our methods do not depend upon the theory of modular forms and are somewhat more transparent.
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Liu, ZG. An Identity of Ramanujan and the Representation of Integers as Sums of Triangular Numbers. The Ramanujan Journal 7, 407–434 (2003). https://doi.org/10.1023/B:RAMA.0000012425.42327.ae
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DOI: https://doi.org/10.1023/B:RAMA.0000012425.42327.ae