Arithmetic Properties of -Regular Bipartitions

Abstract

Let $B_{k,\ell }(n)$ denote the number of $(k,\ell )$-regular bipartitions of $n$. Employing both the theory of modular forms and some elementary methods, we systematically study the arithmetic properties of $B_{3,\ell }(n)$ and $B_{5,\ell }(n)$. In particular, we confirm all the conjectures proposed by Dou [‘Congruences for (3,11)-regular bipartitions modulo 11’, Ramanujan J.40, 535–540].

Ramanujan type of congruences modulo m for (l, m)-regular bipartitions

“…From equation (6.16), (6.17), (6.18), (6. 19) and (6.20), we have q 8 S 4 − q 7 S 3 + 2q 6 S 2 − 3q 5 S + 5q 4…”

“…The reader may refer to [14,16,17,19] for related works. Recently, Adiaga and Ranganatha [1] proved infinite families of congruences modulo 3 for B 3,7 (n).…”

“…Put k = 8 in (7 19),. we can easily find that C(8) ≡ 2 (mod 17) and c(8) ≡ 13 (mod 17), therefore This completes the proof of equation (7.16), (7.17) and (7.18), follow from (7.25) and (7.26). )…”

Arithmetic properties of cubic and overcubic partition pairs

“…We now recall some of the definitions and results from [16,17] which will be used to prove our results. Also, see [18]. For a positive integer M , let R(M ) be the set of integer sequences r = (r δ ) δ|M indexed by the positive divisors of M .…”

Arithmetic Properties of 3-Regular Partitions in Three Colours

“…Arithmetic properties of -regular partition functions have been studied by many authors, including [3,[5][6][7][11][12][13][14].…”