Congruences modulo powers of $3$ for generalized Frobenius partitions $CΨ_{6,0}$

TL;DR

This work derives congruences modulo powers of $3$ for generalized Frobenius partitions, focusing on the $cψ_{6,0}(n)$ case. It builds a bridge between $CΨ_{6,3}$ and $CΨ_{6,0}$ via an Atkin-Lehner involution and analyzes the resulting generating functions through eta-quotients and Jacobi theta identities within a vector-valued modular form framework. The main result shows that if $n,α≥1$ satisfy $2n ≡ -1$ (mod $3^α$), then $cψ_{6,0}(n) ≡ 0$ (mod $3^{⌊α/2⌋+2}$), extending 3-adic congruence phenomena for higher colored Frobenius partitions. The approach demonstrates the effective use of Atkin-Lehner involutions to transfer congruence information between related partition families, with potential implications for broader classes of generalized Frobenius partitions.

Abstract

In 1984, Andrews introduced the family of partition functions $cφ_k(n)$, the number of generalized Frobenius partitions of $n$ with $k$ colors. We have proved a conjecture on congruences modulo powers of 3 for $cφ_6(n)$ before. In this paper, we establish a connection between $CΨ_{6,3}$ and $CΨ_{6,0}$ via an Atkin-Lehner involution and prove congruences modulo powers of 3 for $cψ_{6,0}(n)$.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• proof
• Definition 2.7