Arithmetic properties of the $\ell$-regular partitions

TL;DR

This paper investigates the arithmetic properties of $ ext{p}$-dissection identities related to Ramanujan's theta functions and derives numerous congruences modulo 2 for various $ ext{l}$-regular partition functions, expanding classical results.

Contribution

It introduces new congruences for $ ext{l}$-regular partition functions using $p$-dissection identities and classical Ramanujan congruences, revealing novel arithmetic properties.

Findings

Infinite families of modulo 2 congruences for $ ext{l}$-regular partitions.

New congruences for $ ext{l}$-regular partition functions for specific $ ext{l}$ values.

Extension of classical Ramanujan congruences to $ ext{l}$-regular partitions.

Abstract

For a given prime $p$, we study the properties of the $p$-dissection identities of Ramanujan's theta functions $\psi(q)$ and $f(-q)$, respectively. Then as applications, we find many infinite family of congruences modulo 2 for some $\ell$-regular partition functions, especially, for $\ell=2,4,5,8,13,16$. Moreover, based on the classical congruences for $p(n)$ given by Ramanujan, we obtain many more congruences for some $\ell$-regular partition functions.