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 $M_k\left(n\right)$ 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 $M_k\left(n\right)$ 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.