The bestknown properties and formulas of partitions
Simple values at zero and infinity
The partition functions and are defined for zero and infinity values of argument by the following rules:
Specific values for specialized variables
The following table represents the values of the partitions and for and some powers of 10:
Analyticity
The partition functions and are non‐analytical functions that are defined only for integers.
Periodicity
The partition functions and do not have periodicity.
Parity and symmetry
The partition functions and do not have symmetry.
Series representations
The partition functions and have the following series representations:
where is a special case of a generalized Kloosterman sum:
Asymptotic series expansions
The partition functions and have the following asymptotic series expansions:
Generating functions
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 , .
Identities
The partition functions and satisfy numerous identities, for example:
Complex characteristics
As real valued functions, the partitions and have the following complex characteristics:
Summation
There exist just a few formulas including finite and infinite summation of partitions, for example:
Inequalities
The partitions and satisfy various inequalities, for example:
Congruence properties
The partitions have the following congruence properties:
Zeros
The and partitions have the following unique zeros:
