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Euler-type Recurrence Relations for Partition Functions with Congruence Conditions

This paper derives infinite families of Euler-type recurrence relations for partition functions with specific congruence conditions using generalized Dedekind eta functions and Rankin-Cohen brackets, while also establishing a Rademacher-type formula and a Ramanujan-type congruence as key corollaries.

Original authors: Wissam Raji, Hasan Saad

Published 2026-07-31

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Wissam Raji, Hasan Saad

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

Imagine you are standing in a vast, magical library where the books aren't made of paper, but of numbers. In this library, there is a special section dedicated to "partitions." A partition is simply a way of breaking a whole number down into smaller pieces that add up to the original. For example, the number 4 can be split into 4, or 3+1, or 2+2, or 2+1+1, or 1+1+1+1. Mathematicians have been obsessed with counting how many different ways you can do this for any given number. It's like asking, "How many unique ways can I build a tower of blocks using exactly 100 bricks?"

For over a century, mathematicians have discovered that these counts follow hidden, rhythmic patterns, almost like a secret code. One of the most famous patterns, discovered by Leonhard Euler, acts like a recipe: to find the number of ways to partition a number, you add and subtract the counts of smaller numbers in a very specific, repeating sequence. This paper dives into a more complex version of that recipe. Instead of allowing any block size, imagine a rule that says you can only use blocks that are a certain size, or a size that is a specific "distance" away from a multiple of a big number. The authors are trying to find the new, secret recipes that govern these restricted building games. They use powerful tools from the world of "modular forms"—which are like mathematical shapes that look the same no matter how you stretch or twist them in a specific way—to crack the code.

The New Recipe for Restricted Towers

The authors, Wissam Raji and Hasan Saad, are tackling a specific puzzle: What happens if you are only allowed to build your number towers using blocks that fit a certain "congruence" rule? In math-speak, this means the block sizes must leave a specific remainder when divided by a number $\delta$ . For instance, if $\delta = 5$ , you might only be allowed to use blocks of size 1, 4, 5, 6, 9, 10, etc. (numbers that are 0, 1, or 4 when divided by 5).

The paper's main discovery is that even with these strict rules, there is still a beautiful, infinite family of "Euler-type" recipes. Just like Euler's original recipe told you how to find the total number of partitions by adding and subtracting previous answers, these new recipes do the same thing for the restricted towers. However, the new recipes are much more complex. They don't just add and subtract; they also mix in "divisor sums" (adding up the factors of a number) and special numbers that come from the Fourier coefficients of "cusp forms."

To put it simply, the authors found a way to translate the problem of counting these restricted towers into a language of waves and shapes. They used tools called "generalized Dedekind eta functions" (which are like mathematical engines that generate these partition numbers) and "Rankin–Cohen brackets" (which are like a special blender that mixes two mathematical functions together to create a new one). By blending these functions, they proved that the number of ways to build these restricted towers is directly linked to the behavior of these complex wave-like shapes.

A Concrete Example: The Case of Five

To show their method works, the authors zoomed in on a specific case: $\delta = 5$ and $g = 1$ . This is the rule where you can only use blocks that are 0, 1, or 4 modulo 5. They derived a very specific, explicit formula (Theorem 1.1) for this scenario. This formula says that to find the number of ways to build a tower of size $n$ , you need to:

Look at previous tower counts (using the same pentagonal number pattern as Euler).
Add in some divisor sums (calculating the sum of cubes of the factors of $n$ ).
Subtract a specific number $b(n)$ , which comes from a unique "cusp form" of weight 4 and level 5.

This isn't just a theoretical curiosity; it leads to a "Ramanujan-type congruence." This means the authors proved that for every number $n$ , the mysterious number $b(n)$ is always equal to a specific combination of divisor sums, modulo 13. It's like discovering that no matter how you build your tower, the leftover crumbs always add up to a multiple of 13.

The "Rademacher" Treasure Map

Beyond just finding recipes, the paper also provides a "Rademacher-type formula." If the recurrence relations are like a step-by-step instruction manual, this formula is like a treasure map that lets you calculate the answer directly without having to count every single step before it. It involves "Kloosterman sums" (which are like complex riddles involving remainders) and "Bessel functions" (which describe wave patterns). The authors showed that by treating their generating function as a "Poincaré series" (a type of infinite sum that averages out over a group of symmetries), they could write down an exact formula for the number of partitions. This formula involves summing up contributions from all the "cusps" (the edges or corners of the mathematical shape they are working with), weighted by these Kloosterman sums and Bessel functions.

How They Did It

The authors didn't just guess these formulas; they proved them rigorously. They started by showing that the function generating these partition numbers is a "modular form" of a specific weight. Then, they used a technique called "unfolding" to compute the "Petersson inner product" (a way of measuring how much two mathematical functions overlap). By comparing the "Fourier coefficients" (the numbers in the sequence) of their generated function with a basis of known functions (Eisenstein series and cusp forms), they were able to isolate the exact recurrence relation.

In short, this paper takes a classic problem in number theory—counting ways to break numbers apart—and upgrades it for a more complex set of rules. It proves that even with these new restrictions, the universe of numbers still sings in a predictable, rhythmic pattern, and it provides the exact sheet music (the recurrence relations and formulas) to read that song. The results are not just suggestions or simulations; they are mathematical proofs, establishing a firm connection between partition counting, divisor sums, and the deep, wave-like structures of modular forms.

Technical Summary: Euler-Type Recurrence Relations for Partition Functions with Congruence Conditions

Problem Statement
This paper investigates partition functions $p_{\delta,g}(n)$ , which count the number of partitions of an integer $n$ into parts congruent to $0$ or $\pm g \pmod \delta$ , where $\delta \ge 5$ and $g \in \{1, \dots, \delta-1\}$ with $g \neq \delta/2$ . While Euler's classical recurrence for the standard partition function $p(n)$ involves pentagonal numbers and alternating signs, this work seeks to generalize such recurrence relations to $p_{\delta,g}(n)$ . Specifically, the authors aim to derive infinite families of Euler-type recurrences that express $p_{\delta,g}(n)$ in terms of divisor sums and Fourier coefficients of cusp forms, extending previous work on $t$ -color and $t$ -regular partitions.

Methodology
The authors employ the theory of generalized Dedekind eta functions and Rankin–Cohen brackets to establish their results. The core methodology proceeds as follows:

Generating Functions and Modularity: The generating function for $p_{\delta,g}(n)$ is identified as a generalized Dedekind eta product, $\eta^*_{\delta,g}(\tau)$ , which behaves as a modular form of weight $1/2$ with a specific multiplier system on the congruence subgroup $\pm \Gamma_1(\delta)$ .
Rankin–Cohen Brackets: The authors construct a modular form of weight $2\nu$ (for $\nu \ge 1$ ) by taking the $\nu$ -th Rankin–Cohen bracket of $\eta^*_{\delta,g}$ and its inverse, $[\eta^*_{\delta,g}, (\eta^*_{\delta,g})^{-1}]_\nu$ . The Fourier coefficients of this bracket are shown to correspond to the left-hand side of the desired recurrence relation.
Decomposition into Basis Elements: The constructed modular form is decomposed into a linear combination of Eisenstein series and cusp forms on $\Gamma_1(\delta)$ $Γ_{1} (δ)$ .
- The Eisenstein components are expressed in terms of divisor sums involving Dirichlet characters and generalized Bernoulli numbers.
- The cusp form components are analyzed using the Petersson inner product.
Unfolding and Poincaré Series: To compute the Petersson inner product between the Rankin–Cohen bracket and a cusp form, the authors express the inverse of the eta function as a Poincaré series (specifically, a harmonic Maass form Poincaré series). By "unfolding" the integral defining the inner product, they relate the coefficients to twisted Dirichlet $L$ -series and Kloosterman sums.
Rademacher-Type Formula: The explicit description of the generating function as a Poincaré series allows for the derivation of a Rademacher-type exact formula for $p_{\delta,g}(n)$ involving Bessel functions and Kloosterman sums.

Key Contributions and Results

Generalized Euler Recurrences (Theorem 1.2): The primary result is an infinite family of recurrence relations for $p_{\delta,g}(n)$ $p_{δ, g} (n)$ . For $\nu \ge 2$ $ν \geq 2$ , the relation equates a weighted sum of partition values (involving a polynomial $g_\nu$ $g_{ν}$ and pentagonal-like numbers $\omega(\delta, g; k)$ $ω (δ, g; k)$ ) to a combination of:
- Divisor sums $\sigma_{2\nu-1}(\chi, n)$ weighted by coefficients derived from a matrix inversion involving generalized Bernoulli numbers.
- A trace term involving the Fourier coefficients of cusp forms, expressed as infinite sums of values of twisted Dirichlet $L$ -series.
Explicit Case for $\delta=5$ (Theorem 1.1): The paper provides a concrete recurrence for $p_{5,1}(n)$ $p_{5, 1} (n)$ (partitions into parts $\equiv 0, \pm 1 \pmod 5$ $\equiv 0, \pm 1 (mod 5)$ ) with $\nu=2$ $ν = 2$ . This formula expresses $p_{5,1}(n)$ $p_{5, 1} (n)$ using divisor sums and the Fourier coefficients $b(n)$ $b (n)$ of the unique cusp form of weight 4 and level 5.
- Corollary: This specific case yields a Ramanujan-type congruence modulo 13 relating the coefficients $b(n)$ to divisor sums.
Rademacher-Type Formula (Theorem 1.3): The authors derive an exact formula for $p_{\delta,g}(n)$ as an infinite series involving Kloosterman sums $K_c(\rho, \delta, m, n)$ and modified Bessel functions $I_{3/2}$ . This formula is obtained by explicitly analyzing the principal parts of the generating function at the cusps of $\Gamma_1(\delta)$ .
Non-vanishing Condition: The paper includes a proof that the leading coefficient of the recurrence relation, $g_\nu(\delta, g; n, 0)$ , is non-zero for all $n \ge 1$ , ensuring the relation is indeed a valid recurrence.

Significance
The paper establishes that the recurrence relations for partition functions with congruence conditions are not isolated phenomena but are special cases of a broader framework involving modular forms of various weights. By utilizing Rankin–Cohen brackets and the theory of harmonic Maass forms, the authors successfully link combinatorial partition identities to deep arithmetic objects, including Hecke traces, twisted Dirichlet series, and Kloosterman sums. The work extends the scope of Euler-type recurrences beyond the classical partition function to a wide class of restricted partition functions, providing both recursive algorithms for computation and exact asymptotic/explicit formulas.

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