Arithmetic Properties For (r, s)-Regular Partition Functions With Distinct Parts

Abstract

For any relatively prime integers r and s, let ar,s(n) denote the number of (r, s)- regular partitions of a positive integer of n into distinct parts. Prasad and Prasad (2018) proved many infinite families of congruences modulo 2 for a3,5(n). In this paper, we establish families of congruences modulo 2 and 4 for ar,s(n) with (r, s) ∈ {(2, 5), (2, 7), (4, 5), (4, 9)}. For example, we show that for all β ≥ 0 and n ≥ 0, we have a2,5 4 · 5 2β+1n + 37 · 5 2β − 1 6 ≡ 0 (mod 4) .

Keywords

• regular partition
• distinct parts
• q-series identities
• ramanujan’s theta-functions