Extensions of some results of Jovovic and Dhar

We look at extensions of formulas given by Jovovic and recently proved by Dhar on integer partitions where the smallest part occurs at least $m$ times and on integer partitions with fixed differences between the largest and smallest parts where the smallest part occurs at least $k$ times. Our results extend Dhar’s results for the $m=2$ and $k=1$ cases to the general cases for arbitrary $m$ and $k$. We also look at analogous results for overpartitions and $\mathrm{\ell }$-regular partitions.