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Extensions of some results of Jovovic and Dhar

Pankaj Jyoti Mahanta, Manjil P. Saikia

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.

Extensions of some results of Jovovic and Dhar

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 times and on integer partitions with fixed differences between the largest and smallest parts where the smallest part occurs at least times. Our results extend Dhar's results for the and cases to the general cases for arbitrary and . We also look at analogous results for overpartitions and -regular partitions.
Paper Structure (8 sections, 17 theorems, 46 equations)

This paper contains 8 sections, 17 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['G1']}
  3. Generating Function of
  4. Generating Functions of
  5. Other Classes of Partitions
  6. Overpatitions
  7. -regular partitions
  8. Concluding Remarks

Key Result

Proposition 1.1

For all natural numbers $n$, we have

Theorems & Definitions (25)

  • Proposition 1.1: Formula 1, dhar2021proofs
  • Proposition 1.2: Formula 2, dhar2021proofs
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Proposition 1.9
  • Remark 1.10
  • ...and 15 more