Divisibility properties of weighted $k$ regular partitions

TL;DR

The paper introduces a generalized, weighted family of k-regular partitions with generating function $\sum c_{k,r1,r2}(n) q^n = \prod_{n\ge1} \frac{(1 - q^{nk})^{r1}}{(1 - q^n)^{r2}}$, extending classical k-regular partitions. It develops an eta-quotient/modular-forms framework to construct cusp forms modulo primes and analyzes their Hecke translates, enabling Ramanujan-type congruences and divisibility results. A main achievement is proving a density-one type result for primes: for $r1=r$, $r2=Mr$ with $M$ odd, there exists a positive-density set of primes $\ell$ such that $c_{p,r,Mr}( (dmn\ell - r(p-M))/24 ) \equiv 0 \pmod{m}$ for all $n$ coprime to $\ell$, with corollaries including the classical $b_p(n)$ congruences. The work extends Zheng–Zhao-type density phenomena to broader weighted partition functions, using careful analysis of cusp orders and vanishing under Hecke operators, even when the level grows unbounded.

Abstract

We study a generalized class of weighted $k$-regular partitions defined by \[ \sum_{n=0}^{\infty} c_{k, r_1, r_2}(n) q^n = \prod_{n=1}^{\infty} \frac{(1 - q^{nk})^{r_1}}{(1 - q^n)^{r_2}}, \] which extends the classical $k$-regular partition function $b_k(n)$. We establish new infinite families of Ramanujan-type congruences, divisibility results, and positive-density prime sets for which $c_{k, r_1, r_2}(n)$ vanishes modulo a given prime.

Theorems & Definitions (6)

• Theorem 1.1
• Corollary 1.2
• Theorem 2.1: Gordon–Hughes–Newman
• Theorem 2.2: Ligozat
• Theorem 2.3: J.-P. Serre
• Proposition 2.4