Recursive formulas for sums of squares and sums of triangular numbers

TL;DR

This paper derives recursive formulas for sums of squares and triangular numbers using divisor functions, leading to various mathematical consequences including positivity results for coefficients of infinite products.

Contribution

It introduces new recursive formulas linking sums of squares and triangular numbers to divisor functions, with multiple applications and implications.

Findings

Recursive formulas for sums of squares and triangular numbers.

Connections between these sums and divisor functions.

Positivity results for coefficients of certain infinite products.

Abstract

We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of the coefficients of some infinite products.