### 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) |
---|---|

Pages (from-to) | 191-197 |

Number of pages | 7 |

Journal | Australasian Journal of Combinatorics |

Volume | 60 |

Issue number | 2 |

State | Published - Jan 1 2014 |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics

### Cite this

*Australasian Journal of Combinatorics*,

*60*(2), 191-197.

}

*Australasian Journal of Combinatorics*, vol. 60, no. 2, pp. 191-197.

**Elementary proofs of congruences for the cubic and overcubic partition functions.** / Sellers, James A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Elementary proofs of congruences for the cubic and overcubic partition functions

AU - Sellers, James A.

PY - 2014/1/1

Y1 - 2014/1/1

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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=84923554585&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84923554585&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84923554585

VL - 60

SP - 191

EP - 197

JO - Australasian Journal of Combinatorics

JF - Australasian Journal of Combinatorics

SN - 1034-4942

IS - 2

ER -