An identity of Ramanujan and the representation of integers as sums of triangular numbers

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 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.