Elementary Proofs and Generalizations of Recent Congruences of Thejitha and Fathima

TL;DR

The paper studies Ramanujan-type congruences for $a_k(n)$, the number of partitions with even parts monochromatic and odd parts colored by $k$ colors, with emphasis on $k=5$. It provides purely elementary proofs via generating-function manipulations and $q$-series dissections, avoiding modular form methods. Beyond Thejitha–Fathima's two congruences for $a_5(n)$, the authors derive infinite families, showing for all $j\ge 0$ and $n\ge 0$, $a_{5j+5}(5n+3)\equiv 0 \pmod{5}$ and analogous mod-3 families for several $a_k(n)$, supported by internal congruences. The results broaden the scope of partition-congruence methods, linking to the overpartition function $a_2(n)$ and other $a_k(n)$, and provide a toolkit for deriving further congruences via elementary $q$-series techniques.

Abstract

Motivated by recent work of Hirschhorn and the author, Thejitha and Fathima recently considered arithmetic properties satisfied by the function $a_5(n)$ which counts the number of integer partitions of weight $n$ wherein even parts come in only one color (i.e., they are monochromatic), while the odd parts may appear in one of five colors. They proved two sets of Ramanujan--like congruences satisfied by $a_5(n)$, relying heavily on modular forms. In this note, we prove their results via purely elementary means, utilizing generating function manipulations and elementary $q$-series dissections. We then extensively generalize these two sets of congruences to infinite families of divisibility properties in which the results of Thejitha and Fathima are specific instances.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof : Proof of Theorem \ref{['thm:TF_mod5']}
• Corollary 3.1