Another approach to the Truncated Pentagonal Number Theorem

Recently, in three separate papers, Andrews and Merca, Yee, and Kolitsch and Burnette proved some results concerning the Truncated Pentagonal Number Theorem. In this paper, a family of generating functions that keeps track of the number of different parts in a partition will be studied. This family of generating functions can be used to get the earlier results on the Truncated Pentagonal Number Theorem as well as several other results.