[Paper Review] Elementary proof of congruences modulo 25 for broken $k$-diamond partitions

Shane Chern, Dazhao Tang|arXiv (Cornell University)|Jul 5, 2018

Advanced Mathematical Identities16 references3 citations

TL;DR

This paper provides an elementary proof of an infinite family of congruences modulo 25 for broken $k$-diamond partitions, specifically $\Delta_k(125n + 99) \equiv 0 \pmod{25}$ when $k \equiv 62 \pmod{125}$, using $q$-series identities, modular forms-inspired relations, and recurrence structures in $P(\alpha, \beta)$ functions, avoiding advanced modular form theory. The key contribution is a self-contained, elementary derivation of a result previously proven using modular forms.

ABSTRACT

Let $Δ_{k}(n)$ denote the number of $k$-broken diamond partitions of $n$. Quite recently, the second author proved an infinite family of congruences modulo 25 for $Δ_{k}(n)$ with the help of modular forms. In this paper, we aim to provide an elementary proof of this result.

Motivation & Objective

• To provide an elementary proof of an infinite family of congruences modulo 25 for broken $k$-diamond partitions, previously established using modular forms.
• To replace the analytic modular form approach with algebraic $q$-series identities and recurrence relations.
• To demonstrate that deep arithmetic properties of $\Delta_k(n)$ can be derived using only elementary $q$-series manipulations and functional equations.
• To extend the toolkit of elementary methods in partition congruences beyond modulo 5 to higher powers like 25.

Proposed method

• Defining $P(\alpha, \beta) = x^{\alpha+2\beta}y^{2\alpha-\beta} + (-1)^{\alpha+\beta}q^{2\alpha}/(x^{\alpha+2\beta}y^{2\alpha-\beta})$ with $x = 1/R(q)$, $y = 1/R(q^2)$, and $K = E_2E_5^5/(E_1E_{10}^5)$.
• Establishing recurrence relations for $P(\alpha, \beta)$: $P(\alpha, \beta+1) = \frac{4q}{K}P(\alpha, \beta) + P(\alpha, \beta-1)$, and similar forms for $P(\alpha+2, 0)$ and $P(\alpha+2, -1)$.
• Using the 5-dissection identity for $1/E_1$ in terms of $R(q^5)$ and $E_5, E_{25}$ to express generating functions in terms of $K$ and $q$.
• Applying the recurrence and initial conditions to expand and simplify $\sum_{n=0}^\infty b(5n+4)q^n$ as a polynomial in $K$ and $q$, then reducing modulo 25.
• Connecting the resulting expression to the generating function of $\Delta_k(n)$ via a transformation involving $E_1, E_2, E_5, E_{10}$, and showing that only terms with $q^{5n+4}$ survive modulo 25.
• Deriving the key congruence $\sum_{n=0}^\infty \tilde{c}(5n+4)q^n \equiv 10 \frac{E_5^2 E_{10}^2}{E_1^5 E_2} \pmod{25}$, which implies $c(125n+99) \equiv 0 \pmod{25}$.

Experimental results

Research questions

• RQ1 Can the infinite family of congruences $\Delta_k(125n + 99) \equiv 0 \pmod{25}$ for $k \equiv 62 \pmod{125}$ be proven without modular forms?
• RQ2 What elementary $q$-series identities and recurrence structures can replicate results previously obtained via modular form theory?
• RQ3 How can the function $P(\alpha, \beta)$ be used to systematically reduce generating functions modulo 25?
• RQ4 Is there a way to isolate the $q^{5n+4}$-indexed coefficients in a generating function and show their vanishing modulo 25 through algebraic manipulation?

Key findings

• The paper establishes $\Delta_k(125n + 99) \equiv 0 \pmod{25}$ for all $n \geq 0$ when $k \equiv 62 \pmod{125}$, providing an elementary proof of a result originally proven using modular forms.
• The generating function $\sum_{n=0}^\infty \tilde{c}(5n+4)q^n$ is shown to be congruent to $10 \frac{E_5^2 E_{10}^2}{E_1^5 E_2} \pmod{25}$, which contains no $q^{5n+3}$ terms, implying the vanishing of $c(125n+99)$ modulo 25.
• The recurrence relations for $P(\alpha, \beta)$, initialized from $K = E_2E_5^5/(E_1E_{10}^5)$, allow full reduction of higher-order $q$-series expressions into polynomials in $K$ and $q$.
• The identity $\sum_{n=0}^\infty b(5n+4)q^n = 35K^4 + 280K^3q + 1905K^2q^2 + 1760Kq^3 + 13825q^4 - \frac{7040q^5}{K} + \cdots$ is derived and reduced modulo 25 to confirm the congruence.
• The proof avoids modular forms entirely, relying only on $q$-series identities, recurrence relations, and algebraic simplifications, offering a new elementary route to partition congruences modulo higher powers of 5.