# Elementary proofs of congruences for the cubic and overcubic partition functions

James A. Sellers

Research output: Contribution to journalArticle

3 Citations (Scopus)

### Abstract

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.

In this note, we use elementary means to prove functional equations satisfied by the generating functions for a(n) and ā(n), respectively. These lead to new representations of these generating functions as products of terms involving Ramanujan’s ψand φ functions. In the process, we are able to prove the congruences mentioned above as well as numerous arithmetic properties satisfied by ā(n) modulo small powers of 2.

In 2010, Hei-Chi Chan introduced the cubic partition function a(n) in connection with Ramanujan’s cubic continued fraction. Chan proved that (Formula Presented).

which clearly implies that, for all n ≥ 0, a(3n + 2) ≡ 0 (mod 3).

In the same year, Byungchan Kim introduced the overcubic partition function ā(n). Using modular forms, Kim proved that (Formula Presented).

Original language English (US) 191-197 7 Australasian Journal of Combinatorics 60 2 Published - Jan 1 2014

### Fingerprint

Partition Function
Congruence
Generating Function
Imply
Elementary Functions
Modular Forms
Continued fraction
Functional equation
Modulo
Term

### All Science Journal Classification (ASJC) codes

• Discrete Mathematics and Combinatorics

### Cite this

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abstract = "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.In this note, we use elementary means to prove functional equations satisfied by the generating functions for a(n) and ā(n), respectively. These lead to new representations of these generating functions as products of terms involving Ramanujan’s ψand φ functions. In the process, we are able to prove the congruences mentioned above as well as numerous arithmetic properties satisfied by ā(n) modulo small powers of 2.In 2010, Hei-Chi Chan introduced the cubic partition function a(n) in connection with Ramanujan’s cubic continued fraction. Chan proved that (Formula Presented).which clearly implies that, for all n ≥ 0, a(3n + 2) ≡ 0 (mod 3).In the same year, Byungchan Kim introduced the overcubic partition function ā(n). Using modular forms, Kim proved that (Formula Presented).",
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In: Australasian Journal of Combinatorics, Vol. 60, No. 2, 01.01.2014, p. 191-197.

Research output: Contribution to journalArticle

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N2 - 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.In this note, we use elementary means to prove functional equations satisfied by the generating functions for a(n) and ā(n), respectively. These lead to new representations of these generating functions as products of terms involving Ramanujan’s ψand φ functions. In the process, we are able to prove the congruences mentioned above as well as numerous arithmetic properties satisfied by ā(n) modulo small powers of 2.In 2010, Hei-Chi Chan introduced the cubic partition function a(n) in connection with Ramanujan’s cubic continued fraction. Chan proved that (Formula Presented).which clearly implies that, for all n ≥ 0, a(3n + 2) ≡ 0 (mod 3).In the same year, Byungchan Kim introduced the overcubic partition function ā(n). Using modular forms, Kim proved that (Formula Presented).

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