We find certain congruences satisfied by A`(n) for`= 4,8 and 9, where A`(n) denotes the number of `-regular overpartitions of n. We next prove several congruences for po(n) modulo 8 and 16, where po(n) denotes the number of overpartitions of n into odd parts. We use arithmetic properties of modular forms and eta quotients to study distribution of Andrews.
Radu's algorithm on modular forms to derive certain congruences satisfied by cubic partition pairs, overcubic partition pairs and Andrews' integer partitions with even parts under odd parts. For example, `-regular overpartitions, Andrews singular overpartitions, odd-partition overpartitions, cubic and overcubic partition pairs, and Andrews integer partitions with even parts under odd parts are some of the partition functions studied by many mathematicians. We find certain congruences satisfied by A`(n) for ` = 4,8 and 9, where A`(n) denotes the number of `-regular overpartitions of n.
Our proofs use fundamental properties of the q-series and various dissection formulas of Ramanujan's theta functions. We prove some congruences for po(n) modulo 8 and 16, where po(n) denotes the number of odd-part overparts.
Modular forms
The cusps of a subgroup Γ0 ≤Γ are the equivalence classes of i∞ (also known as "the cusp at infinity") under the influence of Γ0. The condition (2) of the above definition means that f(z) is meromorphic to the thresholds of Γ0. We say that f(z) is a holomorphic modular (resp. cusp) form if f(z) is holomorphic on H and is holomorphic (resp. vanishes) on the cusp of Γ0.
We say that f(z) is a weakly holomorphic modular form if its poles (if any) are supported at the vertices of Γ0. We denote the complex vector space of modular forms (or peak forms) with weight k in relation to Γ0(N) by Mk Γ0(N). We recall the definition of the Hecke operators on spaces of integer-weighted modular forms.
Let A denote the subset of integer weighted modular forms inMk(Γ0(N), χ) whose Fourier coefficients are in OK, the ring of algebraic integers in a number field K. ForA3(n) andA4(n), Shen finds [53 ] some explicit results about the generating function dissections and derive some congruences modulo 3, 6 and 24.
Congruences for A 4 (n)
Congruences for A 8 (n)
Congruences for A 2` (n) and A 4` (n)
We now use induction on α to complete the proof. We then proceed with similar steps as shown in the proof of Theorem 2.2 and derive. 2.62) Similarly, if we extract the terms containing qpn from both sides of (2.62) and then replace qp byq, we get the result. We then extract the terms containing qpn from both sides of the above congruences, and see that and (2.66) is true when α is replaced by α+ 1.
Congruences for A 9 (n)
Using (1.6) on the generating function of A9(3n) in (2.84) and then applying the binomial theorem, we obtain. By extracting the terms containing qpn on both sides of (2.101) and then replacing qp byq, we obtain. Suppose we have α > 0 for someone. by replacing qp byq we derive that. 2.107) Again we obtain, by extracting the terms containing qpn on both sides of (2.107) and then replacing qp byq.
This feature has been highlighted in a number of recent papers, but in contexts very different from overpartitioning. They establish a number of arithmetic results, including several Ramanujan-type congruences satisfied by po(n), and some easy-to-formulate characterizations of po(n) modulo small powers of 2. The following two Ramanujan-type congruences for example, can easily be seen from one of their key positions.
Using the elementary theory of modular forms, he further proves infinitely many congruences for po(n) modulo 32 and 64. 3.7) The first two congruences hold for all nonnegative integers satisfying 8n6≡ −7 (mod p1).
Some generating functions for p o (n)
Congruences for p o (n)
Taking out the terms containing q2n+1, and then using the binomial theorem, we obtain modulo 8,. 3.25) Now, taking out the terms containing q4n and q4n+1, respectively, we have. If we extract the terms containing q3n+1 and q3n+2 from (3.28), the following two Ramanujan-type congruences can be easily obtained.
Two new dissection formulas
Infinite families of congruences for p o (n)
Similarly, if we extract the terms containing qpn from both sides of (3.43) and then replace qp byq, we get the result. We then extract the terms containing qpn on both sides of the above congruence, and see that (3.45) is true when α is replaced by α+ 1. Therefore, we extract the terms containing qpn+17p. 3.51) Similarly, if we extract the terms containing qpn from both sides of (3.51) and then replace qp byq, we get the result.
1The content of this chapter was published in The Ramanujan J.(2018) and some parts are revised. Chen, Hirschhorn and Sellers [17] later showed that Andrews' congruences modulo 3 are two examples of an infinite family of congruences modulo 3 that hold for the function C3,1(n). In a very recent work, Naika and Gireesh [40] prove that two congruences for C3,1(n) modulo 36 proved by Ahmed and Baruah hold for modulo 72.
Proof of Naika and Gireesh’s conjecture
Modularity of eta-quotient
Proof of Theorem 4.3 and Theorem 4.4
We note that all other values of d will give the following four values of D listed in the table. Inspired by Chan's work, Zhao and Zhong [57] studied the partition cubic pair function b(n) defined by.
Proof of a conjecture of Lin
For a positive integer M, let R(M) be the set of integer seriesr = (rδ)δ|M, indexed by the positive divisors of M. For any integer s, let [s]m be the residual class of s in Zm denote , and Sm the set of squares in Z∗m. To apply Lemma 5.2 we use the following result, which gives a complete set of representatives of the double cosets in Γ0(N)\Γ/Γ∞.
Ramanujan-type congruences for overcubic partition pairs
Distribution of a(n) and b(n)
We complete the proof of the theorem by proving the same for the partition function a(n). In a recent paper, Andrews [5] studied the partition function EO(n) which counts the number of partitions of n where each even part is smaller than each odd part. He denoted by EO(n), the number of partitions counted by EO(n) in which only the largest even part occurs an odd number of times.
Andrews proved that the partition function EO(n) has the following generating function [5, Eqn. 6.1) In the same paper, he proposed to undertake a wider investigation of the properties of EO(n).
Infinite families of congruences for EO(n)
Below we present the evidence Ahlgren gave us via e-mail. It turns out that the linear combination of the forms Fj is the eigenform of all Hecke operators. Since Fj are supported by different classes of coefficients, it follows that Fj are eigenforms of all Hecke operators.
Ramanujan-type congruences for EO(n)
Parity of EO(2n)
For any arithmetic progression r (mod t) there are infinitely many integers N ≡r (mod t) for which c(N) is even. For any arithmetic progression r (mod t), there are infinitely many integers M ≡ r (mod t) for which c(M) is odd, provided such an M exists. Moreover, if there exists an M ≡ r (mod t) for which c(M) is odd, then the smallest M is less than Cr,t for.
EO(n) is almost always even
Now, using Corollary 1.14 and proceeding similarly as in the proof of Theorem 4.3, we arrive at the desired result due to (6.34). The web of modularity: arithmetic of the coefficients of modular forms and q-series, volume 102 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004.
