  Regarding Euler's odd-strict theorem, which is the most basic partition
identity, A refinement was done by Sylvester, and it was generalized by
Bessenrodt to the r-regular and r-class regular cases. In this paper, we focus
on the periodicity of the exponent seen on the r-regular side in Bessenrodt's
generalization, and further generalize from that point of view. At the end of
this paper, we also introduce a partition identity when extended without regard
to the periodicity of the exponent.
