Ramanujan's theta functions and linear combinations of four triangular numbers

TL;DR

This paper explores formulas and conjectures related to representing integers as sums of four triangular numbers, utilizing Ramanujan's theta functions to analyze these representations.

Contribution

It introduces new formulas and conjectures for counting representations of integers as sums of four triangular numbers using Ramanujan's theta functions.

Findings

Derived multiple formulas for $t(a,b,c,d;n)$

Proposed several conjectures on representations

Connected theta functions with triangular number sums

Abstract

Let $\Bbb Z$ and $\Bbb Z^+$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb Z^+$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x+1)/2+by(y+1)/2+cz(z+1)/2+dw(w+1)/2$ $(x,y,z,w\in\Bbb Z)$. In this paper, by using Ramanujan's theta functions $\varphi(q)$ and $\psi(q)$ we present many formulas and conjectures on $t(a,b,c,d;n)$.