Arithmetic properties of $2^α-$Regular overpartition pairs
Hemanthkumar B., Sumanth Bharadwaj H. S
TL;DR
This work extends Ramanujan-type congruences to the 2-power regular overpartition pairs $\overline{B}_{2^{\alpha}}(n)$ by developing a generating-function framework in terms of Ramanujan theta functions $\varphi$ and $\psi$, and the Huffing operator $H$. Using precise 2-adic valuation analyses of the associated coefficient matrices, the authors establish infinite families of congruences modulo powers of 2, including $\overline{B}_{2^{\alpha}}(2^{\alpha}(n+1)) \equiv 0 \pmod{2^{3}}$ and $\overline{B}_{2^{\alpha}}(2^{\alpha+\beta+1}(n+1)) \equiv 0 \pmod{2^{3\beta+5}}$, as well as more refined results for arguments of the form $2^{\alpha}(2n+1)$ and $2^{\alpha+1}(n+1)$ involving $\sigma(n)$ and cubic polynomials in $n+1$. The paper also derives higher-modulus congruences and corollaries based on prime-factor constraints, enriching the understanding of 2-adic properties of partition-type functions. The methods combine generating-function decompositions, operator techniques, and detailed arithmetic of auxiliary sequences to yield new Ramanujan-type congruences with potential implications for related partition statistics.
Abstract
Recently, several mathematicians have investigated various partition functions with the goal of discovering Ramanujan-type congruences. One such function is $\overline{B}_{2^α}(n)$, which represents the number of $2^α-$regular overpartition pairs of $n$. In this context, we establish Ramanujan-type congruences modulo powers of $2$ for this function. For instance, we prove that \begin{equation*} \overline{B}_{2^α}(2^{α+β+1}(n+1)) \equiv 0\pmod{2^{3β+5}} \end{equation*} for all $n, β\geq 0,\, α\in \mathbb{N}$.