Right edge rates of the zeros of $\widetilde{\Xi}_n$ and $\widetilde{\Lambda}_n$

1. Problem Statement

The paper investigates the asymptotic behavior of the largest zeros (the "right edge") of two specific families of rescaled even polynomials, $\tilde{\Xi}_n(x)$ and $\tilde{\Lambda}_n(x)$, defined for $n \geq 2$.

These polynomials are derived from the families $\Xi_n$ and $\Lambda_n$ introduced in prior work [1], which are connected to integral representations of normalized values of the Dirichlet beta function $\beta(2n)$ and the Riemann zeta function $\zeta(2n+1)$.

• Definitions:
  • $\tilde{\Xi}_n(x) := \Xi_n(\sqrt{x})$
  • $\tilde{\Lambda}_n(x) := \Lambda_n(\sqrt{x})$
• Properties: Both families are polynomials of degree $n-1$ with all zeros being real, simple, and contained strictly within the interval $(0, 1)$.
• The Gap: While the global distribution of the zeros for both families is known to converge to the same limiting distribution function $F(x)$, the behavior of the extreme right zeros (those closest to 1) was not fully characterized. Specifically, the paper asks: Do the largest zeros of these two families approach the endpoint 1 at the same exponential rate?

2. Methodology

The author employs an elementary but rigorous approach based on the explicit representation of the polynomials in terms of Eulerian polynomials and a transformation mapping the interval $(0, 1)$ to the negative real axis.

A. Transformation to Negative Zeros

The core of the method relies on a change of variables (Lemma 1.1) that maps a zero $x \in (0, 1)$ to a negative zero $z = -a$ (where $a \in (0, 1)$) of an associated Eulerian polynomial:
$$z = -\frac{1-\sqrt{x}}{1+\sqrt{x}} \implies x = \left(\frac{1-a}{1+a}\right)^2$$
Crucially, as $a \to 0^+$, the distance to the right endpoint behaves as:
$$1 - x \sim 4a$$
Thus, determining the rate at which $x \to 1$ is equivalent to determining the rate at which the smallest magnitude negative zero $a$ of the Eulerian polynomial approaches 0.

B. Connection to Eulerian Polynomials

The polynomials $\Xi_n$ and $\Lambda_n$ are explicitly expressed using Eulerian polynomials of Type B ($B_m$) and Type A ($A_m$):

• $\Xi_n$ relates to $B_{2n-1}$.
• $\Lambda_n$ relates to $A_{2n}$.
  The largest zero of $\tilde{\Xi}_n$ corresponds to the negative zero of $B_{2n-1}$ closest to the origin, and similarly for $\tilde{\Lambda}_n$ and $A_{2n}$.

C. Coefficient Estimation (Lemma 1.2)

The author utilizes a bound for the smallest positive root $a_1$ of a polynomial $P(z) = 1 + c_1 z + \dots + c_d z^d$ with positive coefficients and real negative roots:
$$\frac{1}{c_1} \leq a_1 \leq \frac{d}{c_1}$$
This implies that $\log a_1 = -\log c_1 + O(\log d)$. Therefore, the exponential rate of the root is determined entirely by the first non-constant coefficient ($c_1$) of the polynomial.

D. Calculation of Leading Coefficients

The paper calculates the first non-constant coefficients for the relevant Eulerian polynomials:

• Type A ($A_{2n}$): The coefficient of $z$ is $c^{(A)}_{n,1} = 2^{2n} - 2n - 1 \sim 4^n$.
• Type B ($B_{2n-1}$): The coefficient of $z$ is $c^{(B)}_{n,1} = 3^{2n-1} - 2n \sim 9^n$ (specifically $3^{2n-1}$).

3. Key Contributions and Results

Main Theorem (Theorem 2.1)

The paper proves that the two families exhibit distinct exponential rates of convergence for their largest zeros toward 1.

1. For $\tilde{\Lambda}_n$ (Type A):
   $$\lim_{n \to \infty} \frac{1}{n-1} \log\left(1 - x^{(\Lambda)}_{n-1,n}\right) = -\log 4$$
   This implies $1 - x^{(\Lambda)}_{n-1,n} \approx 4^{-(n-1)}$.

2. For $\tilde{\Xi}_n$ (Type B):
   $$\lim_{n \to \infty} \frac{1}{n-1} \log\left(1 - x^{(\Xi)}_{n-1,n}\right) = -\log 9$$
   This implies $1 - x^{(\Xi)}_{n-1,n} \approx 9^{-(n-1)}$.

Numerical Verification

Section 4 provides a numerical table confirming these limits.

• The normalized log-gap for $\tilde{\Lambda}_n$ converges rapidly to $-\log 4 \approx -1.386$.
• The normalized log-gap for $\tilde{\Xi}_n$ converges to $-\log 9 \approx -2.197$.
  The data shows that while the global distribution is identical, the "edge" behavior is fundamentally different.

4. Significance and Interpretation

• Microscopic vs. Macroscopic Behavior: The results highlight a phenomenon where the macroscopic limiting distribution (which describes the bulk of the zeros) fails to capture the microscopic behavior of the extreme zeros. Two sequences can share the same global limit law but possess different local scaling laws at the boundaries.
• Role of Eulerian Coefficients: The paper establishes a direct link between the asymptotic rate of the extreme zeros and the growth rate of the first non-constant coefficient of the underlying Eulerian polynomials.
  • The factor $4^n$ in Type A leads to a rate of $4^{-n}$.
  • The factor $9^n$ in Type B leads to a rate of $9^{-n}$.
• Methodological Simplicity: The proof is notable for being "elementary." It avoids complex analysis or heavy asymptotic machinery, relying instead on algebraic transformations and basic bounds on polynomial roots. This makes the result accessible and highlights the power of coefficient analysis in root distribution problems.

In summary, the paper resolves the asymptotic behavior of the rightmost zeros for these specific polynomial families, demonstrating that despite sharing a global limit, their boundary behaviors are governed by different exponential scales ($4^{-n}$ vs $9^{-n}$) derived from their respective Eulerian polynomial structures.