# Number Identities and Integer Partitions

## TL;DR

This paper explores number identities derived from the triple product identity, applying them to integer partitions and divisor functions, and establishing recursive formulas and theta function frameworks.

## Contribution

It introduces a framework using the triple product identity to derive new number identities, recursive formulas, and theta functions related to modular sets.

## Key findings

- Polygonal number identities from the triple product identity
- Recursive formulas for partition counts with modular parts
- A recursive formula for the sum of divisors function

## Abstract

Using a specific form of the triple product identity, polygonal number identities are stated. Further number identities are examined that can be considered identities related to modular sets of numbers. The identities can be used to give results on integer partitions with parts from numbers in modular arithmetic progression. This includes recursive formulas for the number of partitions using these modular parts. The triple product identity can derive further recursive formulas. Additionally, there is a recursive formula for the related sum of divisors function. The specific triple product identity provides a framework to examine all the identities and can be used to define related theta functions.