Overpartition Function Research Articles

Lovejoy and Osburn’s Overpartitions

In 2008, Lovejoy and Osburn defined the generating function for .In 2009, Byungchan Kim defined the generating function for .This paper shows how to discuss the generating functions for and . Byungchan Kim also defined with increasing relation and overpartition congruences mod 4,8 and 64. In 2006, Berndt found the relation has two values with certain restrictions and various formulae by the common term .This paper shows how to prove the four Theorems about overpartitions modulo 8.These Theorems satisfy the arithmetic properties of the overpartition function modulo 8.

Lovejoy and Osburn’s Overpartitions

In 2008, Lovejoy and Osburn defined the generating function for .In 2009, Byungchan Kim defined the generating function for .This paper shows how to discuss the generating functions for and . Byungchan Kim also defined with increasing relation and overpartition congruences mod 4,8 and 64. In 2006, Berndt found the relation has two values with certain restrictions and various formulae by the common term .This paper shows how to prove the four Theorems about overpartitions modulo 8.These Theorems satisfy the arithmetic properties of the overpartition function modulo 8.

HECKE-TYPE CONGRUENCES FOR TWO SMALLEST PARTS FUNCTIONS

We prove infinitely many congruences modulo 3, 5, and powers of 2 for the overpartition function [Formula: see text] and two smallest parts functions: [Formula: see text] for overpartitions and M2spt(n) for partitions without repeated odd parts. These resemble the Hecke-type congruences found by Atkin for the partition function p(n) in 1966 and Garvan for the smallest parts function spt(n) in 2010. The proofs depend on congruences between the generating functions for [Formula: see text], [Formula: see text], and M2spt(n) and eigenforms for the half-integral weight Hecke operator T(ℓ2).

Two truncated identities of Gauss

Two new expansions for partial sums of Gaussʼ triangular and square numbers series are given. As a consequence, we derive a family of inequalities for the overpartition function p¯(n) and for the partition function pod(n) counting the partitions of n with distinct odd parts. Some further inequalities for variations of partition function are proposed as conjectures.

On the number of partitions of n into k different parts

We study the number of partitions of n into k different parts by constructing a generating function. As an application, we will prove mysterious identities involving convolution of divisor functions and a sum over partitions. By using a congruence property of the overpartition function, we investigate values of a certain convolution sum of two divisor functions modulo 8.

Rademacher-type formulas for restricted partition and overpartition functions

A collection of Hardy-Ramanujan-Rademacher type formulas for restricted partition and overpartition functions is presented, framed by several biographical anecdotes.

Weakly holomorphic modular forms of half-integral weight with nonvanishing constant terms modulo $\ell $

Let l be a prime and λ, j ≥ 0 be an integer. Suppose that f(z) = Σ n a(n)q n is a weakly holomorphic modular form of weight λ + 1/2 and that a(0) ≢ 0 (mod l). We prove that if the coefficients of f(z) are not well-distributed modulo l j , then λ = 0 or 1 (mod l―1/2). This implies that, under the additional restriction α(0) ≢ 0 (mod l), the following conjecture of Balog, Darmon and Ono is true: if the coefficients of a modular form of weight λ+1/2 are almost (but not all) divisible by l, then either λ ≡ 0 (mod l―1/2) or λ ≡ 1 (mod l―1/2). We also prove that if λ ≢ 0 and 1 (mod l―1/2), then there does not exist an integer β, 0 < β < l, such that α(ln + β) ≡ 0 (mod l) for every nonnegative integer n. As an application, we study congruences for the values of the overpartition function.

A short note on the overpartition function

In this brief note, we give a combinatorial proof of a variation of Gauss’s q -binomial theorem, and we determine arithmetic properties of the overpartition function modulo 8.

Partitions and overpartitions with attached parts

We show how to interpret a certain q-series as a generating function for overpartitions with attached parts. A number of families of partition theorems follow as corollaries.

The overpartition function modulo small powers of 2

In a recent paper, Fortin et al. (Jagged Partitions, arXivmath. CO/0310079, 2003) found congruences modulo powers of 2 for the values of the overpartition function p ¯ ( n ) in arithmetic progressions. The moduli for these congruences ranged as high as 64. This note shows that p ¯ ( n ) ≡ 0 ( mod 64 ) for a set of integers of arithmetic density 1.

Overpartitions and real quadratic fields

It is shown that counting certain differences of overpartition functions is equivalent to counting elements of a given norm in appropriate real quadratic fields.