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Eta-Quotient Representations for a Three Parameter Family of Modular Functions Associated with the Rogers-Ramanujan Continued Fraction

Motivated by the 2021 work of Chern and Tang on two-parameter modular functions, this paper constructs a three-parameter extension associated with the Rogers-Ramanujan continued fraction, deriving recursive eta-quotient representations that facilitate dissection formulas, including applications to overpartitions with restricted odd differences.

Original authors: Bishnu Paudel, James A. Sellers, Haiyang Wang

Published 2026-09-14

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CC BY 4.0

Original authors: Bishnu Paudel, James A. Sellers, Haiyang Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of mathematics, there exists a branch dedicated to the study of patterns that repeat with perfect, rhythmic precision. These patterns, known as modular functions, behave like intricate maps that reveal deep connections between numbers, shapes, and the very structure of the universe. For over a century, mathematicians have been fascinated by a specific, mysterious sequence called the Rogers–Ramanujan continued fraction. Imagine this as a never-ending chain of fractions, where each new link depends on the one before it, creating a structure that seems simple on the surface but hides profound complexity. This chain was first noticed in the late 19th century and later rediscovered by the legendary Indian mathematician Srinivasa Ramanujan, who saw in it a wealth of hidden relationships. These relationships are not just abstract curiosities; they serve as powerful tools for solving difficult problems in number theory, particularly those involving how numbers can be broken down into sums, a field known as partition theory.

Building on recent work by other mathematicians who explored two different variations of this problem, a team of researchers has now expanded the scope of this investigation. They have constructed a new, more flexible framework that allows for three adjustable variables instead of just two. By doing so, they have uncovered a set of rules that describe how these complex mathematical objects change when their parameters are shifted. The researchers proved that these new objects can be expressed using a specific type of formula involving infinite products, which act as a kind of universal language for these patterns. Their work provides a complete map of how these functions behave, offering a powerful new method to dissect and understand complicated number patterns that were previously out of reach.

The story begins with the Rogers–Ramanujan continued fraction, a mathematical expression that looks like a fraction within a fraction, stretching on forever. While it appears to be a simple chain of numbers, it possesses a unique symmetry that allows it to be rewritten in different, equally valid ways. Mathematicians have long known that this fraction is intimately connected to the way integers can be partitioned, or broken down into smaller parts. In 2021, two researchers named Chern and Tang introduced a way to study this fraction by creating two families of related functions, each controlled by two numbers. They discovered that these functions could be described using a specific set of building blocks, which are essentially ratios of infinite products. These building blocks allowed them to break down complex formulas into simpler pieces, a process known as dissection, which is crucial for understanding the properties of certain types of number partitions.

Motivated by this success, the authors of the current paper, Bishnu Paudel, James A. Sellers, and Haiyang Wang, asked a natural question: could this approach be extended to include a third variable? They set out to create a three-parameter family of functions, a more general system that would encompass the previous two-parameter versions as special cases. To do this, they defined a new function that combines the Rogers–Ramanujan fraction and its variations in a specific way, controlled by three integers. This new function, which they call G, acts as a master key. By adjusting the three numbers, one can generate the specific functions studied by Chern and Tang, as well as many new ones that had not been explored before.

The core achievement of this work is the discovery of a set of recurrence relations, which are rules that describe how the value of the function changes when one of the three numbers is increased by one. The researchers proved that if you know the value of the function for a certain set of numbers, you can calculate the value for a nearby set of numbers using a simple formula. These formulas rely on a small collection of specific mathematical constants, which are themselves built from the same infinite products used in the original work. The team also established the starting values for these rules, providing a complete foundation from which any member of this new family can be calculated. This is a significant step forward because it turns a potentially infinite and chaotic set of possibilities into a structured, predictable system.

To ensure their results were correct, the authors had to prove that these new functions behave exactly as they claimed. They demonstrated that these functions are what mathematicians call modular functions, meaning they possess a high degree of symmetry and stability under specific transformations. They showed that these functions are smooth and well-behaved everywhere in the complex plane, except at a few specific points where they have predictable behavior. By carefully analyzing the behavior of these functions at these special points, known as cusps, they were able to confirm that the functions they defined are indeed the ones that satisfy the recurrence relations they proposed. This rigorous verification process ensures that the new formulas are not just guesses, but mathematically certain truths.

The implications of this work extend beyond the immediate formulas. The authors note that their results have already been applied in a separate study to solve a specific problem involving overpartitions, a variation of the standard partition problem where certain parts are allowed to be marked in a specific way. By using the new recurrence relations, researchers were able to derive a five-part dissection formula for these overpartitions, revealing a hidden structure in how these numbers are distributed. This demonstrates the practical utility of the new framework: it provides a tool that can be used to unlock secrets in other areas of mathematics that were previously difficult to access.

The paper concludes by placing this new three-parameter family within the broader context of mathematical research. It serves as a generalization of earlier work, showing that the patterns discovered by Chern and Tang are part of a larger, more comprehensive family of relationships. The authors have provided a complete toolkit, including the rules for moving between different values and the specific starting points needed to begin the calculation. This work does not claim to solve every problem in the field, but it offers a robust and reliable method for exploring a wide range of new mathematical territory. By expanding the parameters from two to three, the researchers have opened a door to a richer understanding of the Rogers–Ramanujan continued fraction and its many connections to the world of numbers.

Technical Summary: Eta-Quotient Representations for a Three-Parameter Family of Modular Functions

Problem Statement
The paper addresses the construction and analysis of modular functions associated with the Rogers–Ramanujan continued fraction, $r(\tau)$ . Building upon the 2021 work of Chern and Tang, who introduced two-parameter families of modular functions ( $P(m,n)$ and $Q(m,n)$ ) and established their representation via eta quotients, this work seeks to generalize these results. The primary objective is to construct a three-parameter family of modular functions, denoted $G(m, n, p)$ , and to derive corresponding recurrence relations that express these functions in terms of specific eta quotients.

Methodology
The authors employ the theory of modular functions on congruence subgroups, specifically $\Gamma_0(30)$ . The methodology proceeds through the following steps:

Definition of the Family: The authors define a base function $w(m, n, p)$ involving powers of $r(\tau)$ , $r(2\tau)$ , and $r(3\tau)$ , and construct the family $G(m, n, p) = w + (-1)^{m+n}w^{-1}$ . This construction generalizes the Chern–Tang families, which are recovered as specific cases of $G$ .
Modular Properties: Using generalized Dedekind eta functions ( $\eta_{N,g}$ ), the authors prove that $G(m, n, p)$ is a modular function on $\Gamma_0(30)$ and is holomorphic on the upper half-plane. This involves analyzing the transformation behavior of $r_j(\tau)$ under the action of $\Gamma_0(30)$ .
Initial Value Verification: To establish the recurrence relations, the authors first verify specific initial values for $G(m, n, p)$ at small integer coordinates. They define auxiliary eta quotients ( $K, S, T, U, V$ ) and prove identities relating $G(0,1,0)$ to $1-V$ and $G(0,0,1)$ to $U-2$ . This verification utilizes the valence formula (Lemma 2.8), comparing the orders of the functions at the cusps of $\Gamma_0(30)$ with their $q$ -expansions at infinity.
Derivation of Recurrences: A fundamental algebraic identity (Lemma 4.1) is established, showing that the product of two $G$ functions can be decomposed into a sum of $G$ functions with shifted parameters. By applying this identity with specific unit shifts and substituting the verified initial values, the authors derive linear recurrence relations for the family.

Key Contributions and Results
The central result of the paper is Theorem 1.1, which provides a complete set of recurrence relations and initial values for the three-parameter family $G(m, n, p)$ :

Recurrence Relations: The paper establishes three independent recurrence relations:
- $G(m + 1, n, p) = 4K^{-1} G(m, n, p) + G(m - 1, n, p)$
- $G(m, n + 1, p) = (1 - V) G(m, n, p) + G(m, n - 1, p)$
- $G(m, n, p + 1) = (U - 2) G(m, n, p) - G(m, n, p - 1)$
  Here, $K, U, V$ are specific eta quotients defined in the text.
Initial Values: The paper provides explicit eta-quotient representations for the base cases (e.g., $G(0,0,0)=2$ , $G(1,0,0)=4K^{-1}$ , etc.), some of which were previously known in the literature (Baruah–Begum, Gugg, Chern–Tang) and others proved within this work.
Algebraic Structure: Corollary 1.3 demonstrates that for any integers $m, n, p$ , the function $G(m, n, p)$ belongs to the ring $\mathbb{Q}[K^{\pm 1}, S, T^{\pm 1}, U, V]$ . This confirms that the entire family can be generated from the initial values using the recurrence relations and the specified eta quotients.

Significance and Applications
The paper positions its results as a direct extension of the work by Chern and Tang. The significance lies in providing a systematic framework for generating eta-quotient representations for a broader class of modular functions associated with the Rogers–Ramanujan continued fraction.

The authors explicitly note that these results serve as a primary tool in separate work (reference [18]) to obtain 5-dissection formulas and congruences modulo 5 for the function $t(n)$ , which counts the number of overpartitions of $n$ with restricted odd differences. The paper does not claim further applications beyond this specific use case in partition theory. The work is presented as a technical advancement in the algebraic manipulation of modular functions, facilitating the derivation of dissection formulas that were previously difficult to obtain.

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