Two conjectures of Andrews, Merca and Yee on truncated theta series

Abstract

In their study of the truncation of Euler's pentagonal number theorem, Andrews and Merca introduced a partition function Mk(n) to count the number of partitions of n in which k is the least integer that is not a part and there are more parts exceeding k than there are below k. In recent years, two conjectures concerning Mk(n) on truncated theta series were posed by Andrews, Merca, and Yee. In this paper, we prove that the two conjectures are true for sufficiently large n whenever k is fixed.