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

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## 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 σ_{11}(*n*), and *t*_{24}(*n*). Our methods do not depend upon the theory of modular forms and are somewhat more transparent.

*n*)-function Jacobi's identity modular forms

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