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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:
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