The partition functions and are defined for zero and infinity values of argument by the following rules:
The following table represents the values of the partitions and for and some powers of 10:
The partition functions and are non‐analytical functions that are defined only for integers.
The partition functions and do not have periodicity.
The partition functions and do not have symmetry.
The partition functions and have the following series representations:
where is a special case of a generalized Kloosterman sum:
The partition functions and have the following asymptotic series expansions:
The partition functions and can be represented as the coefficients of their generating functions:
where is the coefficient of the term in the series expansion around of the function , .
The partition functions and satisfy numerous identities, for example:
As real valued functions, the partitions and have the following complex characteristics:
There exist just a few formulas including finite and infinite summation of partitions, for example:
The partitions and satisfy various inequalities, for example:
The partitions have the following congruence properties:
The and partitions have the following unique zeros:
