Extensions of some results of Jovovic and Dhar

TL;DR

This paper extends existing formulas for integer partitions and overpartitions, generalizing previous results to arbitrary parameters and exploring new cases involving fixed differences and multiplicity conditions.

Contribution

It generalizes formulas for partitions with minimum part multiplicity and fixed differences, also extending results to overpartitions and -regular partitions.

Findings

Extended Dhar's results to arbitrary m and k

Derived new formulas for overpartitions and -regular partitions

Provided comprehensive generalizations of partition formulas

Abstract

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 $\ell$-regular partitions.