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Asymptotic zero distribution of the polynomials $\widetildeΞ_n$

This paper establishes the weak convergence of the empirical zero distributions of the rescaled polynomials $\widetilde{\Xi}_n$ to a deterministic probability measure on $(0,1)$ , providing explicit formulas for its density and distribution function through the analysis of type B Eulerian polynomials and the Stieltjes transform method.

Original authors: Luc Ramsès Talla Waffo

Published 2026-02-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Luc Ramsès Talla Waffo

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 have a magical machine that spits out a different, complex mathematical recipe every time you press a button. These recipes are called polynomials. If you solve these recipes (find the numbers that make them equal zero), you get a list of "roots" or "zeros."

For a long time, mathematicians have been studying a specific family of these recipes, known as Eulerian polynomials. They are like a famous, well-behaved family of numbers. But recently, a new, slightly stranger cousin was discovered: the Type B Eulerian polynomials.

This paper is about a specific transformation of these Type B polynomials. The author, Luc Ramsès Talla Waffo, took these complex recipes, tweaked them slightly (by plugging in square roots), and asked a simple question: "If I keep making these recipes bigger and bigger, where do their zeros end up?"

Here is the story of what he found, explained without the heavy math.

1. The Setup: A Crowd of Dancers

Imagine you have a huge crowd of dancers (the zeros of the polynomial).

For a small recipe (say, $n=10$ ), you have a few dancers scattered around a stage.
As you make the recipe bigger ( $n=100$ , then $n=1,000$ ), you get thousands of dancers.
The author noticed that all these dancers are confined to a specific dance floor: the interval between 0 and 1.

The question is: How are they arranged? Are they standing in a straight line? Are they clumped in the corners? Or are they dancing randomly?

2. The Discovery: The "Density Map"

The author proved that as the number of dancers grows to infinity, they don't just scatter randomly. They settle into a very specific, predictable pattern. He created a "density map" (a mathematical picture) that shows exactly how crowded the dance floor is at any given spot.

Here is the shape of the crowd:

The Left Side (Near 0): The dancers are packed together, but they spread out gently as you move away from the edge. It's like a gentle slope where the crowd gets thinner.
The Right Side (Near 1): This is where it gets wild. The dancers are squeezed incredibly tight against the wall at 1. It's like a mosh pit where everyone is pressed so hard against the barrier that it feels like a solid wall.

3. The Two Different Rules of the Dance

The most fascinating part of the paper is that the dancers behave differently at the two ends of the stage:

At the 0 end: The spacing follows a "square root" rule. If you look at the first few dancers, the distance between them grows slowly. It's a predictable, gentle curve.
At the 1 end: The spacing follows a "logarithmic" rule. This is much more extreme. The dancers get closer and closer to the wall at 1 at an exponential rate.
- Analogy: Imagine walking toward a wall. At the 0 end, you take normal steps. At the 1 end, you are shrinking your steps so fast that you never quite reach the wall, but you get infinitely close to it. The last few dancers are practically glued to the wall.

4. The "Magic Formula"

The author didn't just guess this; he wrote down the exact mathematical formula (the "density function") that describes this crowd.

If you were to take a snapshot of the crowd with $n=1,000,000$ dancers, the author's formula would predict exactly how many dancers are in any specific slice of the floor.
He also provided a "distribution function," which is like a cumulative scorecard. It tells you: "If you stand at point $x$ on the floor, what percentage of the crowd is to your left?"

5. Why Does This Matter?

You might ask, "Who cares about where zeros of a polynomial stand?"

It's a Universal Pattern: Just as snowflakes have a specific crystal structure, or galaxies form spiral arms, mathematical objects often have hidden, universal shapes when you look at them on a large scale. This paper reveals the "crystal structure" of Type B Eulerian polynomials.
Connecting the Dots: The author shows that these new polynomials are related to older, well-known ones (Type A). It's like discovering that a new species of bird is actually a cousin of a famous eagle, and now we understand its migration patterns by looking at the eagle's.
Predictability: Even though the individual zeros might look chaotic for small numbers, the author proves that for large numbers, the chaos turns into perfect order. This is a beautiful example of how randomness often hides a deterministic law underneath.

The Bottom Line

Think of this paper as a traffic report for a mathematical highway.
The author drove down the road of these polynomials and realized that while the traffic looks messy up close, if you zoom out, you see a perfect, predictable flow. The cars (zeros) are all stuck between mile markers 0 and 1. They are driving normally in the middle, but they are piling up in a massive traffic jam right at the exit (mile 1), while spreading out gently at the entrance (mile 0).

The paper gives us the exact blueprint of this traffic jam, proving that even in the complex world of advanced math, there is a beautiful, orderly rhythm waiting to be discovered.

1. Problem Statement

The paper investigates the asymptotic distribution of the real zeros of a specific family of polynomials, denoted as $\tilde{\Xi}_n(x)$ , defined for $n \geq 2$ . These polynomials are derived from the polynomials $\Xi_n(x)$ introduced in previous work [5], which are expressed in terms of Eulerian polynomials of type B ( $B_m(z)$ ).

The specific object of study is the rescaled polynomial:
$\tilde{\Xi}_n(x) := \Xi_n(\sqrt{x})$
The author aims to determine the limiting behavior of the empirical zero counting measures $\mu_n$ associated with the zeros of $\tilde{\Xi}_n$ in the interval $(0, 1)$ as $n \to \infty$ . Specifically, the goal is to prove weak convergence to a deterministic probability measure $\mu$ and to derive explicit formulas for its density and distribution function.

2. Methodology

The proof strategy relies on a combination of complex analysis, asymptotic analysis of special functions, and the Stieltjes transform method. The key methodological steps include:

Representation and Logarithmic Derivative: The author expresses $\tilde{\Xi}_n(x)$ explicitly using type B Eulerian polynomials. By taking the logarithmic derivative, the problem is transformed into analyzing the ratio of consecutive Eulerian polynomials, $B_{2n}(y)/B_{2n-1}(y)$ , where $y$ is a Möbius transformation of $x$ .
Asymptotics of Type B Eulerian Polynomials:
- Lemma 1.1: Establishes the asymptotic behavior of the series $S_m(x) = \sum_{k \geq 0} (2k+1)^m x^m$ using contour integration and the residue theorem. It proves that $\lim_{m \to \infty} \frac{1}{m} \frac{S_{m+1}(x)}{S_m(x)} = \frac{2}{-\log x}$ .
- Lemma 1.2: Utilizes the identity relating $S_m(x)$ to $B_m(x)$ to derive the ratio asymptotic for Eulerian polynomials: $\lim_{m \to \infty} \frac{1}{m} \frac{B_{m+1}(x)}{B_m(x)} = \frac{2(1-x)}{-\log x}$ .
Stieltjes Transform Approach:
- The normalized logarithmic derivative $s_n(z) = \frac{1}{n-1} \frac{\tilde{\Xi}'_n(z)}{\tilde{\Xi}_n(z)}$ is identified as the Stieltjes transform of the empirical measure $\mu_n$ .
- Using the ratio asymptotics from Lemma 1.2, the author proves that $s_n(z)$ converges locally uniformly to a limiting function $s(z)$ on $\mathbb{C} \setminus [0, 1]$ .
Inversion Formula: The density of the limiting measure $\mu$ is recovered from the imaginary part of the limit function $s(z)$ using the Stieltjes inversion formula: $\rho(x) = -\frac{1}{\pi} \lim_{\epsilon \to 0^+} \Im s(x + i\epsilon)$ .

3. Key Contributions and Results

A. Weak Convergence (Theorem 2.2)

The paper proves that the sequence of empirical measures $\mu_n$ converges weakly to a unique probability measure $\mu$ supported on the interval $(0, 1)$ .

B. Explicit Limiting Density

The density function $\rho(x)$ of the limiting measure is derived explicitly as:
$\rho(x) = \frac{2}{\sqrt{x}(1-x) \left( \log^2 \left( \frac{1-\sqrt{x}}{1+\sqrt{x}} \right) + \pi^2 \right)}, \quad 0 < x < 1$
and $\rho(x) = 0$ otherwise.

Structure: The density exhibits a log-Cauchy structure, reminiscent of limiting distributions for classical Eulerian polynomials.
Singularities: The density diverges at both endpoints:
- Near $x=0$ : $\rho(x) \sim \frac{2}{\pi^2 \sqrt{x}}$ (square-root singularity).
- Near $x=1$ : The divergence is logarithmic, leading to a "pile-up" of zeros.

C. Limiting Distribution Function (Theorem 2.3)

The cumulative distribution function (CDF) $F(x)$ of the limiting measure is given by:
$F(x) = \begin{cases} 0 & x \leq 0 \\ \frac{2}{\pi} \arctan \left( \frac{1}{\pi} \log \frac{1+\sqrt{x}}{1-\sqrt{x}} \right) & 0 < x < 1 \\ 1 & x \geq 1 \end{cases}$
The empirical distribution functions $F_n$ converge pointwise to $F$ at all continuity points.

D. Asymptotic Behavior of Extremal Zeros (Section 3)

The author analyzes the behavior of the ordered zeros $x_{k,n}$ :

Left Edge (near 0): The smallest zeros scale as $O(n^{-2})$ . Specifically, $x_{k,n} \sim \frac{\pi^4}{16} \frac{k^2}{n^2}$ for fixed $k$ . This confirms a previous conjecture that the smallest zero converges to 0.
Right Edge (near 1): The largest zeros approach 1 at an exponential rate. The distance $1 - x_{k,n}$ decays exponentially with $n$ , much faster than any polynomial rate, due to the logarithmic singularity in the density.

4. Significance and Context

Extension of Melotti's Work: The paper extends results by Paul Melotti regarding the root distribution of Type A Eulerian polynomials to Type B Eulerian polynomials.
Analytic Insight: It provides a rigorous analytical framework for understanding the zero distribution of polynomials defined via Type B Eulerian numbers, a topic previously studied only numerically or qualitatively.
Universality: The emergence of a log-Cauchy type density suggests a universality class for the zero distributions of certain families of polynomials related to Eulerian numbers.
Numerical Verification: Section 4 provides numerical experiments confirming that the empirical CDFs converge rapidly to the theoretical limit $F$ , even for moderate values of $n$ .

5. Conclusion

The paper successfully characterizes the asymptotic zero distribution of the polynomials $\tilde{\Xi}_n$ . By leveraging contour integration and the Stieltjes transform, the author derives exact formulas for the limiting density and distribution function. The results reveal a complex structure where zeros cluster heavily near the boundaries of the interval $(0, 1)$ , with distinct scaling behaviors at the left ( $n^{-2}$ ) and right (exponential) edges. This work serves as a significant extension of the theory of Eulerian polynomials from Type A to Type B.