Ramanujan's theta functions and sums of triangular numbers

TL;DR

This paper explores deep connections between Ramanujan's theta functions and the representation counts of integers as sums of triangular numbers and quadratic forms, revealing new relations for specific cases.

Contribution

It establishes novel relations between sums of triangular numbers and quadratic form representations using Ramanujan's theta functions for cases k=3,4.

Findings

Derived formulas linking t(a_1,...,a_k;n) and N(a_1,...,a_k;8n+sum a_i)

Identified specific cases where these relations hold

Enhanced understanding of theta functions in number representations

Abstract

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a_1,a_2,\ldots,a_k,n\in\Bbb N$ let $N(a_1,a_2,\ldots,a_k;n)$ be the number of representations of $n$ by $a_1x_1^2+a_2x_2^2+\cdots+a_kx_k^2$, and let $t(a_1,a_2,\ldots,a_k;n)$ be the number of representations of $n$ by $a_1\frac{x_1(x_1-1)}2+a_2\frac{x_2(x_2-1)}2+\cdots+a_k\frac{x_k(x_k-1)}2$ $(x_1,\ldots,x_k\in\Bbb Z$). In this paper, by using Ramanujan's theta functions $\varphi(q)$ and $\psi(q)$ we reveal many relations between $t(a_1,a_2,\ldots,a_k;n)$ and $N(a_1,a_2,\ldots,a_k;8n+a_1+\cdots+a_k)$ for $k=3,4$.