Congruence Properties Modulo Prime Powers for a Class of Partition Functions

Author(s)

• Matthew Boylan, University of South CarolinaFollow
• Swati .

Document Type

Article

Abstract

Let p be prime, and let p_[1,p](n) denote the function whose generating function is

∏ _n≥1 (1 − qⁿ)⁻¹ (1 − q^pn)⁻¹.

This function and its generalizations p_[c^r,d^m](n) are the subject of study in several recent papers. Let ℓ ≥ 5, let j ≥ 1, and let p ∈ {2, 3, 5}. In this paper, we prove that the generating function for p_[1,p](n) in the progression β_p,ℓ,j modulo ℓ^j with 24β_p,ℓ,j ≡ p + 1 (mod ℓ^j) lies in a Hecke-invariant subspace of type

{ η(Dz)^r η(Dpz)^s F(Dz) : F(z) ∈ M_s(Γ₀(p), χ) }.

for suitable D ≥ 1, s ≥ 0, and character χ. When p ∈ {2, 3, 5}, we use the Hecke-invariance of these subspaces proved in [21] to prove, for distinct primes ℓ and m ≥ 5 and j ≥ 1, congruences of the form

p_[1,p] ( (ℓ^jm^kn + 1) / D ) ≡ 0 (mod ℓ^j).

for all n ≥ 1 with m ∤ n, where k is explicitly computable and depends on the forms in the invariant subspace. Our proofs require adapting and extending analogous level one results on p(n) in [1] and [22] to level p.

Digital Object Identifier (DOI)

https://doi.org/10.1007/s40993-026-00715-4

Publication Info

Published in Research in Number Theory, Volume 12, 2026.

APA Citation

Boylan, M., & Swati. (2026). Congruence properties modulo prime powers for a class of partition functions. Research in Number Theory, 12.https://doi.org/10.1007/s40993-026-00715-4

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