Congruences modulo powers of 2 for a certain partition function

Abstract

We study the divisibility properties of the coefficients c(n) defined byAn analogue of Ramanujan's partition congruences is obtained for certain coefficients c(n) modulo powers of 2. Furthermore, an analogue of the identity that Hardy regarded as Ramanujan's most beautiful is proved.

The partition function modulo prime powers

“…is the subject of study in several recent papers, including [2], [3], [4], [5], [6], [8], [11], [12], [13], [16], [17], and [20]. The papers [3], [4], [8], [11], [12] and [20] focus on arithmetic properties of p [1,2] (n); [5] concerns p [1 2 ,3 2 ] (n); [6] studies p [1 ℓ ,c m ] for c ∈ {2, 3, 5, 7, 11} and certain ℓ, m ≤ 4; [13] and [17] prove Ramanujanstyle congruences for p [1 ℓ ,11 m ] (n) modulo powers of 11 and [2] proves Ramanujan-style congruences for p [1 ℓ ,(d k ) m ] (n) modulo powers of d where d ∈ {5, 7, 11, 13, 17} and k ≥ 1 is an integer. In [16], Sinick classified all primes ℓ and integers N and a with N ≥ 2 such that p [1,N ] (ℓn + a) ≡ 0 (mod ℓ) for all n.…”

“…The second theorem concerns explicit congruences for p(n) modulo prime powers. The Ramanujan congruences [15] assert, for all n, that p(ℓn + a) ≡ 0 (mod ℓ) when (ℓ, a) ∈ {(5, 4), (7,5), (11,6)}. Ramanujan's work inspired substantial subsequent study of congruences for p(n).…”

New analogues of Ramanujan's partition identities

“…The corresponding analogue for c(n) was subsequently established by Chan and Cooper [7]. The result for a(n) is [6].…”

“…(2.14) 7). (2.16) Recall that in each of the above, the expression on the left-hand side is a modular function on Γ 0 ( e) + W e which is now congruent (modulo ) to a modular form on Γ 0 (e), where the weight of this modular form is divisible by − 1.…”

Asymptotic formulas for general colored partition functions

“…By further analogy, Chan and Cooper [4] and Chen [5] considered a special partition function c(n) which is the number of 4-colored partitions of n with two of the colors appearing only in multiplies of 3. The generating function of c(n) is…”