# Ramanujan's theta functions and linear combinations of three triangular numbers

## TL;DR

This paper explores the relationships between Ramanujan's theta functions and the counts of representations of integers as sums of quadratic and triangular forms, providing new formulas and conjectures for specific parameter cases.

## Contribution

It establishes explicit relations between representations by quadratic and triangular forms using Ramanujan's theta functions and derives formulas for various parameter configurations.

## Key findings

- Relations between $t(2,3,3;n)$ and $N(1,3,3;n+1)$
- Formulas for $t(a,3a,4b;n)$, $t(a,7a,4b;n)$, $t(3a,5a,4b;n)$, and $t(a,15a,4b;n)$ under certain conditions
- Numerous conjectures on $t(a,b,c;n)$ for special $(a,b,c)$ values

## Abstract

Let $\Bbb Z$ be the set of integers. For positive integers $a,b,c$ and $n$ let $N(a,b,c;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2$, and let $t(a,b,c;n)$ be the number of representations of $n$ by $ax(x+1)/2+by(y+1)/2+cz(z+1)/2$ $(x,y,z\in\Bbb Z)$. In this paper, by using Ramanujan's theta functions $\varphi(q)$ and $\psi(q)$ we reveal the relation between $t(2,3,3;n)$ and $N(1,3,3;n+1)$, and the relation between $t(1,1,6;n)$ and $N(1,1,3;n+1)$. We also obtain formulas for $t(a,3a,4b;n),$ $t(a,7a,4b;n),t(3a,5a,4b;n)$ and $t(a,15a,4b;n)$ under certain congruence conditions, where $a$ and $b$ are positive odd integers. In addition, we pose many conjectures on $t(a,b,c;n)$ for some special values of $(a,b,c)$.