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Contour Integration and Cyclotomic Apéry-Like Series Involving Generalized Binomial Coefficients

This paper introduces a contour integration method to evaluate cyclotomic Apéry-like series involving generalized binomial coefficients in terms of multiple polylogarithms and zeta values, while also deriving new identities for Fuss-Catalan integrals and posing open questions in the field.

Original authors: Ce Xu

Published 2026-02-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ce Xu

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 a detective trying to solve a massive, infinite puzzle. The puzzle pieces are numbers, but they are arranged in a very specific, tricky pattern involving fractions, factorials, and powers. This paper is about a mathematician named Ce Xu who has developed a new, powerful magnifying glass to look at these patterns and figure out exactly what they add up to.

Here is a simple breakdown of what the paper does, using some everyday analogies.

1. The Mystery: "Apéry-like" Series

In the world of math, there are famous infinite sums called Riemann zeta values. Think of these as the "DNA" of numbers. Some of these sums are easy to solve (like even numbers), but the ones involving odd numbers (like 3, 5, 7) are like locked safes.

In 1979, a mathematician named Roger Apéry cracked the safe for the number 3. He found a special recipe (an infinite series) that equals $\zeta(3)$ . This recipe used something called central binomial coefficients. You can think of these coefficients as a specific type of "mathematical ingredient" that makes the recipe work.

Since then, mathematicians have been trying to find similar recipes for other numbers. These are called "Apéry-like series." They are like variations of Apéry's original recipe, but they are much harder to solve.

2. The New Tool: Contour Integration

Usually, solving these infinite sums is like trying to count every grain of sand on a beach one by one. It takes forever.

The author of this paper uses a technique called Contour Integration.

The Analogy: Imagine you want to know the total weight of all the fish in a huge, dark ocean. Counting them one by one is impossible. Instead, you cast a giant, magical net (a "contour") around the whole area. By looking at the edges of the net and how the water flows through it, you can calculate the total weight of the fish inside without ever seeing a single fish.
In Math: The author draws a giant circle in the complex number plane (a special kind of map for numbers). By analyzing the "poles" (special points where the math gets wild) inside this circle, he can instantly calculate the value of the entire infinite sum.

3. The Ingredients: Generalized Binomial Coefficients

The series in this paper use Generalized Binomial Coefficients.

The Analogy: A standard binomial coefficient is like a recipe for a cake that calls for exactly 2 eggs and 1 cup of flour. A generalized one is like a recipe that says, "Use $\pi$ eggs and $\sqrt{2}$ cups of flour." It's a more flexible, abstract version of the ingredient.
The paper handles these flexible ingredients by using the Gamma function, which is essentially a super-charged version of the factorial function ( $n!$ ) that works for any number, not just whole numbers.

4. The Result: Translating the Code

The main achievement of the paper is that the author took these messy, infinite sums and translated them into a "dictionary" of known mathematical constants.

Instead of leaving the answer as a long, confusing infinite sum, the author showed that these sums are actually made of:

Logarithms (like $\log(2)$ ).
Zeta values (the famous constants).
Polylogarithms (think of these as "super-logarithms," like a Swiss Army knife for complex numbers).
Hurwitz Zeta values (a more specific type of constant).

The Metaphor: It's like taking a complex, encrypted message written in a secret code (the infinite series) and translating it into plain English using a dictionary of standard words (the constants). Now, instead of staring at a confusing string of symbols, we can read the answer clearly.

5. The Side Quest: Fuss-Catalan Numbers

The paper also takes a detour to look at Fuss-Catalan numbers.

The Analogy: If binomial coefficients are like standard Lego bricks, Fuss-Catalan numbers are like a special, curved Lego brick used to build specific shapes (like trees or polygons).
The author used the "magic net" (contour integration) again to find a new way to calculate sums involving these special bricks. This led to discovering new identities (mathematical equalities) that connect different types of these "super-logarithms."

6. The Open Questions

Even with this success, the author admits there are still two locked safes they couldn't crack yet:

Question 1: Can we use this method if the "root of unity" (a specific type of number in the series) isn't a perfect circle point? (Can we use the magic net on a slightly different shape?)
Question 2: Is there a simple formula for a second type of series that appears in the paper? (We know the answer for one specific case, but what about the general case?)

Summary

In short, Ce Xu wrote a paper that says:

"We have a new, powerful way to solve very difficult infinite number puzzles. By using a technique called 'contour integration' (drawing a magic circle), we can translate these messy puzzles into clean, understandable formulas involving famous constants like $\pi$ , $\log(2)$ , and zeta values. We solved many of them, found some new connections between them, and identified two remaining mysteries for future detectives to solve."

This work is significant because it connects different branches of mathematics (like number theory and complex analysis) and provides a toolkit for solving problems that were previously considered too difficult.

1. Problem Statement

The paper addresses the evaluation of a specific class of infinite series known as cyclotomic parametric Ap´ery-like series. These series generalize the famous series used by Roger Ap´ery to prove the irrationality of $\zeta(3)$ . The specific series investigated is defined as:
$S = \sum_{n=-\infty}^{\infty} \frac{4^{n+a}}{(n+a)^q \binom{2n+2a}{n+a}} z^n$
where:

$a \in \mathbb{C} \setminus \mathbb{Z}$ is a complex parameter.
$q \in \mathbb{N}$ is an integer power.
$z$ is an arbitrary root of unity (with the condition $(q, z) \neq (1, 1)$ ).
The binomial coefficient is generalized via the Gamma function: $\binom{a}{b} = \frac{\Gamma(a+1)}{\Gamma(b+1)\Gamma(a-b+1)}$ .

The primary challenge is to express these series in terms of known special functions, specifically multiple polylogarithms, cyclotomic Hurwitz zeta values, Riemann zeta values, and $\log(2)$ .

2. Methodology

The author employs contour integration and residue calculus as the primary analytical tool. The methodology proceeds as follows:

Kernel Construction: A specific kernel function $\Phi(s-a; x)$ is constructed using an extended trigonometric function involving the generalized digamma function $\phi(s; x)$ . This function is designed to have poles at integers with residues related to the series terms.
Contour Integral: The author considers the contour integral over a large circular contour $C_R$ in the complex plane:
$I = \oint_{C_R} \Phi(s-a; x) \frac{\Gamma^2(1+s)}{\Gamma(1+2s)} \frac{4^s}{s^q} \, ds$
Asymptotic Analysis: Using the Legendre duplication formula and asymptotic expansions of the Gamma function ratio, it is proven that the integral vanishes as the radius $R \to \infty$ (for $q \ge 1$ ).
Residue Theorem Application: Since the integral is zero, the sum of residues at all poles inside the contour must be zero. The poles are identified as:
1. $s = n + a$ ( $n \in \mathbb{Z}$ ): Simple poles corresponding to the terms of the target Ap´ery-like series.
2. $s = -n$ ( $n \in \mathbb{N}$ ): Simple poles related to the reciprocal central binomial coefficients.
3. $s = 0$ : A pole of order $q$ .
Expansion Techniques: The author utilizes:
- Exponential Complete Bell Polynomials ( $Y_n$ ) to handle the Taylor expansions of the Gamma function and its inverse near poles.
- Coefficients $C_n$ and $D_n$ : Derived from the expansion of $\Gamma(s+1)e^{\gamma s}$ and its inverse, involving Riemann zeta values.
- Laurent Expansions: To compute residues at the poles, specifically utilizing the expansion of $\Phi(s-a; x)$ which involves cyclotomic Hurwitz zeta functions ( $\text{Li}_p(x; b)$ ).

3. Key Contributions and Results

A. Main Theorem (Theorem 2.1)

The paper establishes a closed-form expression for the cyclotomic parametric Ap´ery-like series. The series is expressed as a linear combination of:

Multiple Polylogarithms ( $\text{Li}_{k}(x)$ ).
Cyclotomic Hurwitz Zeta Functions ( $\text{Li}_p(x; b)$ ).
Riemann Zeta Values ( $\zeta(k)$ ).
Logarithmic terms ( $\log(2)$ ).

The formula involves Bell polynomials $C_n$ and $D_n$ and sums over partitions of the integer $q-1$ .

B. Corollaries and Special Cases

Recovery of Known Results: By setting specific values for parameters (e.g., $x = \pm 1$ , $a = 1/2$ ), the author recovers known results from previous literature (specifically referencing works by Sun, Zhao, and the author's previous work with Rui).
Corollary 2.6: Provides a specific evaluation for the series where $a = 1/2$ , linking it to the standard central binomial coefficient $\binom{2n}{n}$ .
Theorem 2.7: Provides an alternative expression for the difference of cyclotomic Hurwitz zeta functions using derivatives of $\cot(\pi a)$ , offering a different perspective on the terms appearing in the main theorem.

C. Alternative Expression via Fuss-Catalan Numbers (Section 5)

The author derives an alternative expression for a related class of Euler-Ap´ery-like series by integrating the generating function of Fuss-Catalan numbers ( $G_m(x)$ ).

This leads to Theorem 5.2, which expresses series involving $\binom{mn}{n}$ in terms of multiple polylogarithms and logarithms of $G_m(-x)$ .
Identities: By comparing the results from the contour integration method (Section 2) and the generating function method (Section 5), the author establishes new identities for multiple polylogarithm functions. For example, equation (5.11) relates $\text{Li}_2$ terms with different arguments, and equation (5.12) provides a complex identity involving $\text{Li}_{2,1}$ and $\text{Li}_3$ .

4. Significance

Unification: The paper unifies the evaluation of various Ap´ery-like series under a single framework using contour integration, demonstrating the power of this method for series involving generalized binomial coefficients.
Generalization: It extends previous results (which often focused on specific integer parameters or $x=1$ ) to the cyclotomic case (arbitrary roots of unity) and parametric case (arbitrary complex $a$ ).
New Identities: The derivation of identities between multiple polylogarithms (e.g., relating arguments involving $\sqrt{1-x}$ ) provides new tools for simplifying complex expressions in number theory and mathematical physics.
Open Questions: The author explicitly leaves two questions open for future research:
1. Whether similar closed-form formulas exist for the series when $x$ is not a root of unity (but $|x| \le 1$ ).
2. Whether the second parametric series appearing in Theorem 4.1 admits a closed-form expression for $x \neq 1$ in terms of Gamma or Digamma functions.

5. Conclusion

This paper significantly advances the theory of Ap´ery-like series by successfully applying contour integration to a broad class of cyclotomic parametric series. It not only recovers known results but also generates new identities involving multiple polylogarithms and Hurwitz zeta functions, while highlighting the deep connections between combinatorial series, special functions, and complex analysis.