Some congruences for 12-coloured generalized Frobenius partitions

IF 0.7 3区数学 Q2 MATHEMATICS

Proceedings of the Edinburgh Mathematical Society Pub Date : 2024-05-02 DOI:10.1017/s0013091524000294

Su-Ping Cui, Nancy S. S. Gu, Dazhao Tang

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引用次数: 0

Abstract

In his 1984 AMS Memoir, Andrews introduced the family of functions

$c\phi_k(n)$

, the number of k-coloured generalized Frobenius partitions of n. In 2019, Chan, Wang and Yang systematically studied the arithmetic properties of

$\textrm{C}\Phi_k(q)$

for

$2\leq k\leq17$

by utilizing the theory of modular forms, where

$\textrm{C}\Phi_k(q)$

denotes the generating function of

$c\phi_k(n)$

. In this paper, we first establish another expression of

$\textrm{C}\Phi_{12}(q)$

with integer coefficients, then prove some congruences modulo small powers of 3 for

$c\phi_{12}(n)$

by using some parameterized identities of theta functions due to A. Alaca, S. Alaca and Williams. Finally, we conjecture three families of congruences modulo powers of 3 satisfied by

$c\phi_{12}(n)$

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12 色广义弗罗贝尼斯分区的一些全等式

安德鲁斯在1984年的AMS回忆录中介绍了函数$c\phi_k(n)$族，即n的k色广义弗罗贝纽斯分区数。2019年，Chan、Wang和Yang利用模形式理论系统地研究了$\textrm{C}\Phi_k(q)$对于$2\leq k\leq17$的算术性质，其中$\textrm{C}\Phi_k(q)$表示$c\phi_k(n)$的生成函数。本文首先建立了$\textrm{C}\Phi_{12}(q)$ 的另一个整数系数表达式，然后利用阿拉卡（A. Alaca）、阿拉卡（S. Alaca）和威廉姆斯（Williams）的θ函数的一些参数化同调，证明了$c\phi_{12}(n)$ 的一些小幂次同调。最后，我们猜想$c\phi_{12}(n)$ 满足三个 3 的幂的全等族。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Proceedings of the Edinburgh Mathematical Society 数学-数学

CiteScore

1.10

自引率

0.00%

发文量

审稿时长

6 months

期刊介绍： The Edinburgh Mathematical Society was founded in 1883 and over the years, has evolved into the principal society for the promotion of mathematics research in Scotland. The Society has published its Proceedings since 1884. This journal contains research papers on topics in a broad range of pure and applied mathematics, together with a number of topical book reviews.