Arithmetic properties of singular overpartition pairs without multiples of k. Issue 2 (1st July 2021) - Nayaka, S. Shivaprasada; Sreelakshmi, T.K.; Kumar, Santosh
Abstract : Purpose: In this paper, the author defines the function B ¯ i, j δ, k ( n ), the number of singular overpartition pairs of n without multiples of k in which no part is divisible by δ and only parts congruent to ± i, ± j modulo δ may be overlined. Design/methodology/approach: Andrews introduced to combinatorial objects, which he called singular overpartitions and proved that these singular overpartitions depend on two parameters δ and i can be enumerated by the function C ¯ δ, i ( n ), which gives the number of overpartitions of n in which no part divisible by δ and parts ≡ ± i (Mod δ ) may be overlined. Findings: Using classical spirit of q -series techniques, the author obtains congruences modulo 4 for B ¯ 2, 4 8, 3 ( n ), B ¯ 2, 4 8, 5 and B ¯ 2, 4 12, 3 . Originality/value: The results established in this work are extension to those proved in Andrews' singular overpatition pairs of n .