On Glaisher's Partition Theorem

1. Problem Statement and Context

The paper addresses the generalization of classical partition theorems, specifically Euler's Partition Theorem and Glaisher's Partition Theorem.

• Euler's Theorem ($m=2$): States that the number of partitions of $n$ into distinct parts ($A(n)$) equals the number of partitions of $n$ into odd parts ($B(n)$).
• Glaisher's Theorem (General $m$): States that the number of partitions of $n$ into parts repeating at most $m-1$ times ($A_m(n)$) equals the number of partitions of $n$ into parts not divisible by $m$ ($B_m(n)$).

The Specific Gap:
Recent work by Andrews, Kumar, and Yee introduced refined partition functions $C(n)$ and $D(n)$ that satisfy $A(n) = B(n) = C(n+1) = \frac{1}{2}D(n+1)$. Subsequently, Lin and Zang generalized the function $C(n)$ to the case of $m$-regular partitions (parts not divisible by $m$), denoted as $C_m(n)$.

The Research Question:
The authors aim to:

1. Generalize the function $D(n)$ to the case of $m$-regular partitions, denoted as $D_m(n)$.
2. Establish an analogous identity for $m=3$ (and general $m$) relating $C_m(n)$ and $D_m(n)$.
3. Derive new finite and infinite $q$-series identities related to Glaisher's product.

2. Methodology

The authors primarily utilize analytic combinatorics and $q$-series theory (generating functions).

• Generating Functions: The core methodology involves constructing generating functions for the defined partition functions ($C_m(n)$, $D_m(n)$, $E_m(n)$) and manipulating them using standard $q$-series identities.
• Roots of Unity Filter: A critical technique employed is the use of roots of unity ($\zeta_m$) to decompose sums. Specifically, the identity $\frac{1}{m}\sum_{k=0}^{m-1} \zeta_m^{kn}$ is used to isolate terms where indices are multiples of $m$.
• Euler's Identities: The proofs rely heavily on Euler's fundamental $q$-series identities, such as:
  $$\frac{1}{(t;q)_\infty} = \sum_{n=0}^\infty \frac{t^n}{(q;q)_n}$$
• Algebraic Manipulation: The authors perform intricate algebraic expansions, index shifting, and summation interchanges to transform complex generating functions into recognizable forms or to isolate specific coefficients.

3. Key Definitions

To state the results, the authors define the following partition functions:

• $C_m(n)$: The number of partitions of $n$ where the largest part is divisible by $m$ (say $mj$), and all parts $\le j$ repeat less than $m$ times.
• $D_m(n)$: The number of partitions of $n$ into non-negative parts where the smallest part occurs exactly $m$ times, and all other parts repeat less than $m$ times.
• $E_m(n)$: An auxiliary function defined via its generating function $\varepsilon_m(q)$, which accounts for the "error" or remainder when relating $C_m$ and $D_m$.
• $B_m^{(j)}(n)$: Partitions of $n$ into parts not divisible by $m$ with the largest part congruent to $j \pmod m$.

4. Key Results and Contributions

A. Generalization of the $C(n) = \frac{1}{2}D(n)$ Identity

The paper establishes a relationship between $C_m(n)$ and $D_m(n)$ for general $m$.

• Theorem 1.4: For any non-negative integer $n$:
  $$C_m(n) = D_m(n) + \frac{E_m(n)}{m}$$
  where $E_m(n)$ is defined by a specific generating function involving roots of unity.

B. The Case $m=3$ (The Main Theorem)

The authors focus on the $m=3$ case to provide a concrete, elegant identity similar to the $m=2$ case.

• Theorem 1.5: They derive an explicit series expansion for the generating function of the error term $E_3(n)$:
  $$\varepsilon_3(q) = 2 - q - 2q^2 + \sum_{n=2}^\infty (-1)^n \chi(n-1) q^{\binom{n+1}{2} + 1}$$
  where $\chi(n)$ is a character function based on $n \pmod 3$.
• Corollary 1.6: A significant consequence of Theorem 1.5 is that for $n$ not equal to a triangular number plus 1, the error term vanishes, yielding a clean identity:
  $$C_3(n) = \frac{1}{3} D_3(n)$$
  This mirrors the $m=2$ result ($C(n) = \frac{1}{2}D(n)$) but with a specific exception condition.

C. Unified Partition Identity

By combining the generalized Glaisher theorem ($A_m = B_m$) with the new relations for $C_m$ and $D_m$, the authors present a comprehensive identity (Theorem 1.7):
$$A_m(n) = B_m(n) = \sum_{k=1}^{m-2} B_m^{(k)}(n) + C_m(n+1) = \sum_{k=1}^{m-2} B_m^{(k)}(n) + D_m(n+1) + \frac{E_m(n+1)}{m}$$
This unifies the counting of partitions by repetition constraints, divisibility constraints, and the refined structural constraints of $C_m$ and $D_m$.

D. New $q$-Series Identities

The paper provides new finite and infinite $q$-series identities (Theorem 1.8 and Corollary 1.9) that equate a specific sum involving $q$-Pochhammer symbols to the product generating function for $B_m(n)$:
$$1 + \sum_{n=1}^\infty \sum_{j=1}^{m-1} q^{mn-j} \left( \prod_{r=1}^{m-j} (q^r; q^m)_n \right) \left( \prod_{r=m-j+1}^{m-1} (q^r; q^m)_{n-1} \right) = \frac{(q^m; q^m)_\infty}{(q; q)_\infty}$$
This generalizes a known result by Andrews and Bachraoui for $m=2$.

5. Significance

• Extension of Classical Theory: The paper successfully extends the elegant companion functions of Euler's theorem (Andrews-Kumar-Yee) to the broader context of Glaisher's theorem for $m \ge 3$.
• Structural Insight: It reveals that the relationship between "largest part divisible by $m$" ($C_m$) and "smallest part repeated $m$ times" ($D_m$) holds a precise ratio ($1/m$) for almost all $n$ when $m=3$, with the exceptions being explicitly characterized by triangular numbers.
• Analytic Tools: The derivation of the generating function for the error term $E_3(q)$ demonstrates sophisticated manipulation of $q$-series involving roots of unity, providing a template for future generalizations to higher $m$.
• Open Problems: The authors conclude by noting that while the analytic proofs are complete, a bijective proof (a direct combinatorial mapping) for Corollary 1.6 remains an open and interesting challenge.

In summary, this paper bridges the gap between classical partition theorems and modern refined partition identities, providing rigorous analytic proofs for new relationships between partition functions constrained by repetition and divisibility.