Congruences for a class of eta-quotients and their applications

2020/10/04 by Shashika Petta Mestrige, Mestrige, Shashika Petta
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2010.01594

openalex publication_date 2020/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The partition function p_[1^cℓ^d](n) can be defined using the generating function, ∑_n=0^∞p_[1^cℓ^d](n)qⁿ=∏_n=1^∞\dfrac1(1-qⁿ)^c(1-q^{ℓ n})^d. In \citeP, we proved infinite family of congruences for this partition function for ℓ=11. In this paper, we extend the ideas that we have used in \citeP to prove infinite families of congruences for the partition function p_[1^cℓ^d](n) modulo powers of ℓ for any integers c and d, for primes 5≤ ℓ≤ 17. This generalizes Atkin, Gordon and Hughes' congruences for powers of the partition function. The proofs use an explicit basis for the vector space of modular functions of the congruence subgroup Γ₀(ℓ). Finally we used these congruences to prove congruences and incongruences of the generalized Frobenius ℓ-color partitions, ℓ-regular partitions and ℓ-core partitions for ℓ=5,7,11,13 and 17.

Related

• Congruences modulo powers of 11 for some eta-quotients
• Ramanujan congruences for a class of eta quotients
• Congruences like Atkin's for generalized Frobenius partitions
• Congruences like Atkin's for the partition function
• Congruences of the partition function