  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.
