Congruences modulo powers of $3$ for generalized Frobenius partitions $C\Psi_{6,0}$

TL;DR

This paper proves new congruences modulo powers of 3 for generalized Frobenius partition functions, extending previous results and establishing connections via Atkin-Lehner involutions.

Contribution

It introduces novel congruence results for the (6,0)-colored Frobenius partition functions and links their generating functions through an Atkin-Lehner involution.

Findings

Established congruences modulo powers of 3 for cψ_{6,0}(n).

Connected generating functions of cψ_{6,3}(n) and cψ_{6,0}(n) via Atkin-Lehner involution.

Extended previous work on cφ_6(n) to cψ_{6,0}(n).

Abstract

In 1984, Andrews introduced the family of partition functions \(c\phi_k(n)\), which counts the number of generalized Frobenius partitions of \(n\) with \(k\) colors. In previous work, we proved a conjecture on congruences for \(c\phi_6(n)\) modulo powers of 3. In this paper, we consider the \((6,0)\)-colored Frobenius partition functions \(c\psi_{6,0}(n)\). We establish a connection between the generating functions of \(c\psi_{6,3}(n)\) and \(c\psi_{6,0}(n)\) via an Atkin-Lehner involution, and prove congruences modulo powers of 3 for \(c\psi_{6,0}(n)\).