New infinite families of congruences for Andrews' (<i>K,I</i>)-singular overpartitions

Abstract

In a recent work, Andrews dened the singular overpartition functions, denoted by C̄_k;i(n), which count the number of overpartitions of n in which no part is divisible by k and only parts ≡±i (mod k) may be overlined. Moreover, many congruences modulo 3, 9 and congruences modulo powers of 2 for C̄_k;i(n) were discovered by Ahmed and Baruah, Andrews, Chen, Hirschhorn and Sellers, Naika and Gireesh, Shen and Yao for some pair (k; i). In this paper, we proved new innite families of congruences modulo 27 for C̄_3;1(n) and innite families of congruences modulo 4 and 8 for C̄_6;2(n), C̄_12;3(n), C̄_28;7(n).

Mathematics Subject Classication (2010): 11P83, 05A17.

Key words: Congruence, singular overpartition, theta function.