New density results and congruences for Andrews' singular overpartitions

Abstract

Andrews introduced the singular overpartition function C‾k,i(n) which counts the number of overpartitions of n in which no part is divisible by k and only parts ≡±i(modk) may be overlined. In this article, we study the divisibility properties of C‾4k,k(n) and C‾6k,k(n) by arbitrary powers of 2 and 3 for infinite families of k. For an infinite family of k, we prove that C‾4k,k(n) is almost always divisible by arbitrary powers of 2. We also prove that C‾6k,k(n) is almost always divisible by arbitrary powers of 3 for an infinite family of k. Using a result of Ono and Taguchi on nilpotency of Hecke operators, we find infinite families of congruences modulo arbitrary powers of 2 satisfied by C‾4⋅2α,2α(n) and C‾4⋅3⋅2α,3⋅2α(n).