[Paper Review] Arithmetic Identities and Congruences for Partition Triples with 3-cores

Liuquan Wang|arXiv (Cornell University)|Feb 23, 2015

Advanced Mathematical Identities13 references3 citations

TL;DR

This paper establishes new arithmetic identities and congruences for $ B_3(n) $, the number of partition triples of $ n $ with 3-core partitions, using $ q $-series identities and modular forms. It proves three key arithmetic relations between $ B_3(n) $ and $ \omega(n) $, the number of representations of $ n $ as a sum of six squares and three times three squares, including $ \omega(6n+5) = 4B_3(6n+4) $, and derives infinite families of Ramanujan-type congruences for $ B_3(n) $.

ABSTRACT

Let ${{B}_{3}}(n)$ denote the number of partition triples of $n$ where each partition is 3-core. With the help of generating function manipulations, we find several infinite families of arithmetic identities and congruences for ${{B}_{3}}(n)$. Moreover, let $ω(n)$ denote the number of representations of a nonnegative integer $n$ in the form $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+3y_{1}^{2}+3y_{2}^{2}+3y_{3}^{2}$ with ${{x}_{1}},{{x}_{2}},{{x}_{3}},{{y}_{1}},{{y}_{2}},{{y}_{3}}\in \mathbb{Z}.$ We find three arithmetic relations between ${{B}_{3}}(n)$ and $ω(n)$, such as $ω(6n+5)=4{{B}_{3}}(6n+4).$

Motivation & Objective

• Establish new arithmetic identities and congruences for $ B_3(n) $, the count of partition triples of $ n $ where each partition is a 3-core.
• Uncover deep arithmetic connections between $ B_3(n) $ and the number of representations $ \omega(n) $ of $ n $ as a sum of six squares and three times three squares.
• Extend known results on $ a_3(n) $ and $ A_3(n) $ to the case of partition triples ($ B_3(n) $) using $ q $-series and theta function identities.
• Propose open questions on generalizing these results to $ k $-tuples of 3-core partitions for arbitrary $ k $.
• Provide a unified framework for $ A_3^{(k)}(n) $ and $ \omega^{(k)}(n) $, suggesting future research directions.

Proposed method

• The generating function $ \sum_{n=0}^\infty B_3(n) q^n = \frac{f_3^9}{f_1^3} $ is used as the foundation, where $ f_k = (q^k; q^k)_\infty $.
• Employ 2-dissection and 3-dissection identities of $ f_3^3/f_1 $ and related theta functions to decompose the generating function into series with specific congruence conditions.
• Use Ramanujan's theta function identities and modular forms to derive recursive relations and congruences for $ B_3(n) $.
• Extract coefficients from modular forms and $ q $-series expansions to establish exact arithmetic identities between $ B_3(n) $ and $ \omega(n) $.
• Apply generating function manipulation and coefficient extraction techniques to prove identities such as $ \sum_{n=0}^\infty B_3(6n+4) q^n = 24 \frac{f_2^8 f_3^3}{f_1^5} $.
• Compare generating functions of $ B_3(n) $ and $ \omega(n) $ to derive exact arithmetic relations, including $ \omega(6n+5) = 4B_3(6n+4) $.

Experimental results

Research questions

• RQ1 Can infinite families of arithmetic identities and congruences be derived for $ B_3(n) $, the number of partition triples with 3-core partitions?
• RQ2 Are there exact arithmetic relations between $ B_3(n) $ and $ \omega(n) $, the number of representations of $ n $ as $ x_1^2 + x_2^2 + x_3^2 + 3y_1^2 + 3y_2^2 + 3y_3^2 $?
• RQ3 Can the known identities for $ a_3(n) $ and $ A_3(n) $ be generalized to the case of partition triples $ B_3(n) $ using $ q $-series and modular forms?
• RQ4 Is it possible to extend the results to $ k $-tuples of 3-core partitions for arbitrary $ k $, and if so, what identities or relations emerge?
• RQ5 Can new congruences of Ramanujan-type be proven for $ B_3(n) $, such as $ B_3(3^k n + 3^k - 1) \equiv 0 \pmod{3^{2k}} $?

Key findings

• The paper proves that $ \omega(6n+5) = 4B_3(6n+4) $, establishing a precise arithmetic relation between the number of 3-core partition triples and representations as sums of squares.
• An identity $ \omega(12n+2) = 12B_3(6n) $ is derived, linking $ B_3(6n) $ to the number of representations of $ 12n+2 $ in a sum of squares with coefficients.
• Another identity, $ \omega(12n+10) = 6B_3(6n+4) $, is proven, further connecting $ B_3(n) $ to $ \omega(n) $ via generating function comparisons.
• An infinite family of congruences is established: $ B_3(2^{k+1}n + 2^k - 1) \equiv 0 \pmod{\frac{4^{k+1} + (-1)^k}{5}} $ for $ k \geq 1 $ and all $ n \geq 0 $.
• Two Ramanujan-type congruences are proven: $ B_3(30n+10) \equiv B_3(30n+28) \equiv 0 \pmod{120} $, derived from the generating function $ \sum_{n=0}^\infty B_3(6n+4) q^n = 24 \frac{f_2^8 f_3^3}{f_1^5} $.
• An infinite family of congruences is found: $ B_3(3^k n + 3^k - 1) \equiv 0 \pmod{3^{2k}} $, and $ B_3(3^k n + 2\cdot 3^{k-1} - 1) \equiv 0 \pmod{3^{2k-1}} $, for $ k \geq 1 $.