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The pod function and its connection with other partition functions

Hemjyoti Nath

Abstract

The number of partitions of $n$ wherein odd parts are distinct and even parts are unrestricted, often denoted by $pod(n)$. In this paper, we provide linear recurrence relations for $pod(n)$, and the connections of $pod(n)$ with other partition functions.

The pod function and its connection with other partition functions

Abstract

The number of partitions of wherein odd parts are distinct and even parts are unrestricted, often denoted by . In this paper, we provide linear recurrence relations for , and the connections of with other partition functions.
Paper Structure (17 sections, 29 theorems, 132 equations)

This paper contains 17 sections, 29 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Recurrence relations
  4. Connections of $pod(n)$ with other partition functions
  5. Connection with the partition function $ped(n)$
  6. Connection with the partitions into distinct odd parts
  7. Connection with the overpartition and the partition function $A(n)$
  8. Connection with the overpartitions into odd parts
  9. Connection with the ordinary partition function and the partition function $q_{e-o}(n)$
  10. Connection with the partitions into parts congruent to $2\pmod{4}$
  11. Connections with the ordinary partition function
  12. Connections with the cubic partition function
  13. Connection with the partition function $q_{odd}^{"'}(n)$
  14. Relation between the partition functions $pod(n)$, $p(n)$, $a(n)$ and $q_{odd}(n)$
  15. Relation between the partition functions $p(n), \overline{\mathcal{E}\mathcal{O}}(n)$ and $pod(n)$
  16. ...and 2 more sections

Key Result

Theorem 2.1

We have

Theorems & Definitions (53)

  • Theorem 2.1: Euler's Pentagonal Number Theorem 9
  • Theorem 2.2: Jacobi's Identity 9
  • Lemma 2.3: Hirschhorn 2
  • Lemma 2.4: Baruah 3
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • ...and 43 more