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Laurent type expansion of multiple polylogarithms at integer points

This paper investigates the local behavior of multiple polylogarithm functions at integer points by deriving a Laurent-type expansion in the $s$ -aspect, where the coefficients are expressed through regularised values at related integer points.

Original authors: Pawan Singh Mehta, Biswajyoti Saha

Published 2026-01-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Pawan Singh Mehta, Biswajyoti Saha

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 trying to listen to a very complex song, but the speakers are broken at specific, predictable notes. When you hit those notes, the music doesn't just stop; it explodes into static, making it impossible to hear the melody. In the world of advanced mathematics, these "songs" are called Multiple Polylogarithms, and the "broken notes" are specific integer points where the math gets messy and undefined.

This paper, written by Pawan Singh Mehta and Biswajyoti Saha, is like a repair manual for those broken speakers. It teaches us how to tune the music so we can hear what the song should sound like, even at the points where it usually explodes.

Here is the breakdown of their work using everyday analogies:

1. The Problem: The "Exploding" Song

Think of a Multiple Polylogarithm as a giant, infinite recipe for a number. You add up an endless list of ingredients (fractions involving powers of numbers).

The Good News: If you are cooking in a safe kitchen (a specific mathematical area called $U_r(z)$ ), the recipe works perfectly. You get a smooth, delicious number.
The Bad News: If you try to cook at certain specific integer points (like trying to bake a cake at exactly 100 degrees when the oven breaks), the recipe blows up. The numbers go to infinity, and the math breaks down.

For a long time, mathematicians knew where the oven broke, but they didn't have a good way to describe exactly how it broke or what the "flavor" of the explosion was.

2. The Solution: The "Laurent Type Expansion"

The authors introduce a tool called a Laurent type expansion. Imagine you are a mechanic trying to fix a car engine that is making a terrible noise at a specific RPM. Instead of just saying "it's broken," you write down a detailed formula that describes the noise:

Part of the noise is a loud, sharp scream (the "pole" or infinity).
Part of the noise is a steady hum (the "regular" part).
Part of the noise is a subtle vibration (the "correction" terms).

The authors' expansion does exactly this for the math. It takes the exploding function and writes it as a sum of:

The Explosion: A simple fraction that tells you exactly how bad the break is (e.g., $1/(s-1)$ ).
The Regular Part: A smooth, well-behaved function that represents the "true" value of the music underneath the static.
The Coefficients: These are the "secret ingredients" (called regularised values) that the authors calculate. These are the specific numbers that tell you what the function would have been if the explosion hadn't happened.

3. The "Regularisation" Process: Cleaning the Static

How do they find these secret ingredients? They use a process called regularisation.

Imagine you are trying to measure the height of a growing plant, but every time you measure it, a gust of wind knocks the ruler over, adding a random amount of static to your reading.

The Old Way: You just give up because the data is messy.
The Authors' Way: They look at the pattern of the wind. They realize the wind adds a specific, predictable amount of "noise" (like a constant drift or an oscillation). They mathematically subtract that noise from the measurement. What's left is the regularised value—the true height of the plant, stripped of the wind's interference.

In this paper, they apply this "noise subtraction" to the infinite sums. They separate the part that goes to infinity, the part that wiggles around, and the part that stays constant. That constant part is the regularised value, which becomes the new, clean definition of the function at that broken point.

4. The Map: Knowing Where the Breaks Are

The authors didn't just fix one spot; they mapped out the entire landscape.

They created a set of rules (called Theorems 4 and 6) that act like a GPS.
If you tell the GPS your location (the specific integer point and the specific "flavor" of the numbers involved), it tells you:
- Will the function explode here?
- If yes, what does the explosion look like?
- What is the hidden, smooth value underneath?

They found that the "explosion" usually happens along specific lines (hyperplanes) where the numbers add up in a certain way. If you are standing on those lines, the function breaks. If you are off the lines, it's smooth.

5. The Result: A Complete Picture

By the end of the paper, the authors have achieved two main things:

They proved the expansion works: They showed that this "Laurent type expansion" isn't just a guess; it's a mathematically rigorous way to describe the function everywhere, even at the broken points.
They solved the "Holomorphicity" mystery: They answered a tricky question: "Is the function actually smooth at these integer points if we look at it the right way?" They proved that yes, if you use their "regularised" version, the function is actually smooth and well-behaved at these points, provided you are in a specific mathematical zone (called $V_r(z)$ ).

Summary

In simple terms, this paper takes a mathematical object that is known to be "broken" at specific integer points and provides a detailed blueprint for how to fix it. They don't just patch the hole; they write a new version of the function that includes a description of the break and a clean, usable value underneath it. This allows mathematicians to finally "hear" the music clearly, even at the notes that used to cause static.

Technical Summary: Laurent Type Expansion of Multiple Polylogarithms at Integer Points

Problem Statement
The paper addresses the local behavior of multiple polylogarithm functions, denoted as $Li(a_1, \dots, a_r; z_1, \dots, z_r)$ or $Liz(s_1, \dots, s_r)$ , specifically at integer points in the complex domain $\mathbb{C}^r$ . While the convergence and holomorphicity of these functions are well-understood within specific domains (such as $U_r(z)$ ), their behavior at integer points, particularly those outside the standard domain of convergence or where the series diverges, remains delicate. Previous work by the authors [4] established a regularization process for these series and identified a larger domain of convergence, $V_r(z)$ , but did not fully resolve the question of holomorphicity or provide explicit expansions at these integer points. The core problem is to construct explicit Laurent-type expansions for these functions around arbitrary integer points $a \in \mathbb{Z}^r$ and to determine the conditions under which the functions are holomorphic at these points.

Methodology
The authors employ a multi-faceted approach combining regularization techniques, combinatorial identities, and asymptotic analysis:

Regularization and Formal Power Series: The authors utilize a regularization process defined in their previous work [4]. They define the "regularized value" $\ell(a_1, \dots, a_r)_{[k_1, \dots, k_r]}(z)$ as the constant term in the asymptotic expansion of partial sums of the series involving logarithmic terms. This allows them to construct a formal power series (Equation 9) representing the regularized multiple polylogarithm function, $Li^{Reg}_{(z;a)}(s_1, \dots, s_r)$ .
Combinatorial Identities: A crucial step involves the relationship between the multiple polylogarithm function and the multiple polylogarithm-star function ( $Li^\star$ ). The authors use a combinatorial identity (Theorem 5, Equation 19) that expresses the partial sum of the polylogarithm as an alternating sum involving the tail of the star-function and lower-depth polylogarithms.
Asymptotic Expansion of Tails: The paper analyzes the asymptotic behavior of the tails of the star-functions, $Li^\star_{z}(s_1, \dots, s_r)_{\ge N}$ , as $N \to \infty$ . Using translation formulas (Proposition 1) and induction on depth and the integer parameters, the authors derive complete asymptotic expansions for these tails relative to a specific comparison scale $E$ involving powers of $N$ and $\log N$ with coefficients in a specific algebra $\mathbb{C}_z$ .
Matrix Inversion and Rational Functions: To relate the regularized function to the original function, the authors set up systems of linear equations involving the functions at various depths. By expressing these relationships in matrix form (e.g., $V = M^{-1}V_{Reg}$ ), they invert upper triangular matrices with rational function entries. This inversion process yields the explicit coefficients for the Laurent expansion.

Key Contributions and Results

Explicit Laurent-Type Expansion (Theorem 4 & Theorem 6): The primary result is the derivation of an explicit Laurent-type expansion for the multiple polylogarithm function $Liz(s_1, \dots, s_r)$ around any integer point $a \in \mathbb{Z}^r$ .
- For points $a$ within the domain $U_r(z)$ , Theorem 4 provides an expansion where the function is expressed as a sum of regularized functions multiplied by rational functions involving specific hyperplanes $H^{(i)}_j$ .
- Theorem 6 generalizes this to all integer points in $\mathbb{C}^r$ . It establishes that the formal power series of the regularized function converges in a neighborhood of $a$ and extends to a meromorphic function on $\mathbb{C}^r$ . The expansion is given by:
  $Li^{Reg}_{(z;a)}(s) = \sum_{i=0}^r (-1)^i Li_{(z_{i+1}, \dots, z_r)}(s_{i+1}, \dots, s_r) C_i(s_1, \dots, s_i)$
  where $C_i$ are explicitly defined rational functions involving Bernoulli numbers and Eulerian polynomials.
Holomorphicity at Integer Points (Corollary 1): The paper proves that for any integer point $a \in V_r(z)$ , the multiple polylogarithm function $Liz(s)$ is holomorphic at $a$ . Furthermore, the value of the function at $a$ is exactly the regularized value $\ell(a_1, \dots, a_r)_{[0, \dots, 0]}(z)$ . This resolves the delicate question of holomorphicity at these points, which was left open in [4].
Extension to Non-Convergent Points (Corollary 2): The authors show that even for integer points outside $V_r(z)$ , if the parameters $z_i$ satisfy specific conditions (specifically $z_{[1,i]} \neq 1$ for all $i$ ), the function remains holomorphic everywhere in $\mathbb{C}^r$ , and its value at any integer point equals the regularized value.
Asymptotic Analysis of Star-Functions (Proposition 3): The paper provides a complete asymptotic expansion for the tails of the multiple polylogarithm-star functions with variable coefficients, generalizing previous results for single-variable cases.

Significance
The paper claims to extend the work of [6], which provided Laurent expansions for multiple zeta functions, to the broader case of multiple polylogarithm functions with roots of unity as parameters. By establishing the convergence of the regularized power series and providing explicit formulas for the expansion, the authors bridge the gap between the analytic continuation of these functions and their arithmetic values at integer points. The results confirm that the regularization process defined in [4] yields the correct analytic continuation, thereby validating the regularized values as the true values of the meromorphic functions at integer points where the original series may not converge. The work provides a systematic framework for understanding the singularities and local behavior of these functions, which are fundamental in number theory and mathematical physics.