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Arithmetic properties for generalized cubic partitions and overpartitions modulo a prime

September 2024
Aequationes mathematicae

Authors:

University of Virginia

A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number of new congruences akin to the classical Ramanujan-type. We emphasize two methods of proofs, one elementary (relying significantly on functional equations) and the other based on modular forms. We close by proving analogous results for generalized overcubic partitions.

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Aequat. Math.

The Author(s), under exclusive licence to

Springer Nature Switzerland AG 2024

https://doi.org/10.1007/s00010-024-01116-7 Aequationes Mathematicae

Arithmetic properties for generalized cubic partitions and

overpartitions modulo a prime

Tewodros Amdeberhan, James A. Sellers, and Ajit Singh

Abstract. A cubic partition is an integer partition wherein the even parts can appear in two

colors. In this paper, we introduce the notion of generalized cubic partitions and prove a

number of new congruences akin to the classical Ramanujan-type. We emphasize two meth-

ods of proofs, one elementary (relying signiﬁcantly on functional equations) and the other

based on modular forms. We close by proving analogous results for generalized overcubic

partitions.

Mathematics Subject Classiﬁcation. 05A17, 11F03, 11P83.

Keywords. Cubic partitions, Ramanujan congruences, Modular forms.

1. Introduction

A partition λof a positive integer nis a sequence of positive integers λ1≥

λ2≥ ··· ≥ λrsuch that r

i=1 λi=n. The values λ1,λ

2,...,λ

rare called the

parts of λ.Weletp(n) denote the number of partitions of nfor n≥1, and we

deﬁne p(0) := 1.As an example, note that the partitions of n=4are

4,3+1,2+2,2+1+1,1+1+1+1

and this means p(4) = 5.

Throughout this work, we adopt the notations (a;q)∞=j≥0(1 −aqj),

where |q|<1, and fk:= (qk;qk)∞. As was proven by Euler, we know that

P(q):=

n≥0

p(n)qn=

j≥1

1−qj=1

Because the focus of this paper is on divisibility properties of certain partition

functions, we remind the reader of the celebrated Ramanujan congruences for

p(n)[10, p. 210, 230]: For all n≥0,

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Period of the Ikeda type lift for the exceptional group of type E7,3

E_{7,3}

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MATH Z

In the second and the third named authors (Compos Math 152: 223–254, 2016) constructed the Ikeda type lift for the exceptional group E7,3

E_{7,3}

from an elliptic modular cusp form. In this paper, we prove an explicit formula for the period or the Petersson norm of the Ikeda type lift in terms of the product of the special values of the symmetric square L-function of the elliptic modular form. There are similar works done by the first author with his collaborator, but new technical inputs are required and developed to overcome some difficulties coming from the hugeness of E7,3

E_{7,3}

Congruences and recursions for the cubic partition

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Dec 2017
RAMANUJAN J

Let p2(n)

p_2(n)

denote the number of cubic partitions. In this paper, we shall present two new congruences modulo 11 for p2(n).

p_2(n).

We also provide an elementary alternative proof of a congruence established by Chan. Furthermore, we will establish a recursion for p2(n),

p_2(n),

which is a special case of a broader class of recursions.

Ramanujan’s Lost Notebook: Part I

Book

Nov 2005

This volume is the first of approximately four volumes devoted to providing statements, proofs, and discussions of all the claims made by Srinivasa Ramanujan in his lost notebook and all his other manuscripts and letters published with the lost notebook. In addition to the lost notebook, this publication contains copies of unpublished manuscripts in the Oxford library, in particular, his famous unpublished manuscript on the partition and tau-functions; fragments of both published and unpublished papers; miscellaneous sheets; and Ramanujan's letters to G. H. Hardy, written from nursing homes during Ramanujan's final two years in England. This volume contains accounts of 442 entries (counting multiplicities) made by Ramanujan in the aforementioned publication. The present authors have organized these claims into eighteen chapters, containing anywhere from two entries in Chapter 13 to sixty-one entries in Chapter 17. Most of the results contained in Ramanujan's Lost Notebook fall under the purview of q-series. These include mock theta functions, theta functions, partial theta function expansions, false theta functions, identities connected with the Rogers-Fine identity, several results in the theory of partitions, Eisenstein series, modular equations, the Rogers-Ramanujan continued fraction, other q-continued fractions, asymptotic expansions of q-series and q-continued fractions, integrals of theta functions, integrals of q-products, and incomplete elliptic integrals. Other continued fractions, other integrals, infinite series identities, Dirichlet series, approximations, arithmetic functions, numerical calculations, diophantine equations, and elementary mathematics are some of the further topics examined by Ramanujan in his lost notebook.

Elementary proofs of congruences for the cubic and overcubic partition functions

Article

Jan 2014

J.A. Sellers

More recently, Hirschhorn has proven Kim’s generating function result above using elementary generating function methods. Clearly, this generating function result implies that ā(3n + 2) ≡ 0 (mod 6) for all n ≥ 0.

Ramanujan's Lost Notebook

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Jan 2012

Each of these two isolated pages has a connection with Ramanujan’s famous paper in which he gives the first proofs of the congruences p(5n+4)≡0 (mod 5) and p(7n+5)≡0 (mod 7). One of Ramanujan’s proofs hinges upon the beautiful identity \sum_{n=0}^{\infty}p(5n+4)q^n = 5 \frac{(q^5;q^5)_{\i}^5}{(q;q)_{\infty}^6}, \qquad |q|<1, which is given on page 189. We provide a more detailed rendition of the proof given by Ramanujan, as well as a similarly beautiful identity yielding the congruence p(7n+5)≡0 (mod 7). On both pages, Ramanujan examines the more general partition function pr(n) defined by

\frac{1}{(q;q)_{\infty}^r}=\sum_{n=0}^{\infty}p_r(n)q^n, \qquad |q|<1.

In particular, he states new congruences for pr(n).

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Byungchan Kim

Let a ¯(n) be the number of overcubic partitions of n. We show that a ¯(3n+2)≡0(mod3) and give a crank for this Ramanujan-type congruence.

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Hei-Chi Chan

In this paper, we prove an analog of Ramanujan's "Most Beautiful Identity". This analog is closely related to Ramanujan's beautiful results involving the cubic continued fraction.

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