Asymptotics and zero behaviour of geometric polynomials

1. Problem Statement

The paper investigates the asymptotic behavior and the distribution of zeros for two closely related families of polynomials: Geometric polynomials ($P_n(z)$) and Eulerian polynomials ($A_n(z)$).

• Geometric Polynomials: Defined by the generating function $f(z, t) = \frac{1}{1-z(e^t-1)} = \sum_{n=0}^\infty P_n(z) \frac{t^n}{n!}$. They are linked to Stirling numbers of the second kind and ordered set partitions (Fubini numbers).
• Eulerian Polynomials: Defined by $\sum_{k=1}^\infty k^n z^k = \frac{z A_n(z)}{(1+z)^n}$. They count permutations with a specific number of descents.
• The Core Challenge: While the algebraic properties of these polynomials are well-known, their precise asymptotic behavior in the complex plane (specifically outside the interval $[-1, 0]$) and the detailed distribution of their zeros within the interval $(-1, 0)$ require rigorous analysis. The authors aim to derive uniform asymptotic expansions and determine the spacing of zeros in the "bulk" of the interval.

2. Methodology

The authors employ a combination of complex analysis, generating function theory, and asymptotic analysis.

• Generating Functions and Singularities: The analysis relies heavily on the generating functions of $P_n(z)$ and $A_n(z)$. The authors identify the singularities (poles) of these functions in the complex $t$-plane.
  • For $P_n(z)$, the generating function has simple poles at $t_k = \log(\frac{1+z}{z}) + 2\pi k i$.
  • For $z \in (-1, 0)$, the dominant singularities are at $t = \ell(z) \pm \pi i$, where $\ell(z) = \log|\frac{1+z}{z}|$.
• Cauchy's Integral Formula: The coefficients $P_n(z)$ are extracted using Cauchy's integral formula. The authors deform the integration contour to isolate the contribution of the dominant poles, separating the main asymptotic term from the error term.
• Residue Calculus: By calculating residues at the dominant poles, the authors derive explicit asymptotic formulas involving trigonometric functions (specifically cosine) derived from the sum of residues at conjugate poles.
• Symmetry and Orthogonality: The paper utilizes the symmetry property $P_n(z) = (-1)^n (1+z) P_n(-1-z)$ to establish orthogonality relations and interlacing properties of the zeros.
• Transformation to Eulerian Polynomials: A key methodological step is the use of the transformation $P_n(\frac{1}{z}-1) = \frac{A_n(z)}{(z-1)^n}$ to transfer results from Geometric polynomials to Eulerian polynomials, avoiding redundant proofs.

3. Key Contributions and Results

A. Asymptotic Behavior in the Complex Plane

The authors establish the uniform asymptotic behavior of Geometric polynomials on compact subsets of $\mathbb{C} \setminus [-1, 0]$.

• Theorem 3:
  $$\lim_{n \to \infty} \frac{P_n(z) (\log \frac{1+z}{z})^{n+1}}{n!} = \frac{1}{1+z}$$
  This convergence is uniform on compact sets. A similar result is derived for Eulerian polynomials (Theorem 6).
• Implication: This leads to Corollary 1, which describes the limiting distribution of zeros via the logarithmic potential and establishes that the ratio $\frac{n P_{n-1}(z)}{P_n(z)}$ converges to $\log \frac{1+z}{z}$.

B. Asymptotic Behavior in the Interval $(-1, 0)$

Inside the interval where the zeros lie, the behavior is oscillatory.

• Theorem 1: The authors derive a precise asymptotic approximation for $P_n(z)$ in $(-1, 0)$. Let $\ell(z) = \log|\frac{1+z}{z}|$. The polynomial behaves like:
  $$(1+z)P_n(z) \frac{((\ell(z))^2 + \pi^2)^{(n+1)/2}}{n!} \approx J_n(z)$$
  where $J_n(z)$ is a normalized oscillatory term:
  $$J_n(z) = \frac{(\ell(z) + \pi i)^{n+1} + (\ell(z) - \pi i)^{n+1}}{((\ell(z))^2 + \pi^2)^{(n+1)/2}} = 2\cos((n+1)\theta(z))$$
  This approximation holds uniformly on sub-intervals approaching the endpoints $-1$ and $0$ as $n \to \infty$.

C. Zero Distribution (Clock Behavior)

The paper provides a detailed description of the spacing of zeros in the "bulk" of the interval $(-1, 0)$.

• Theorem 2: The zeros $z_{j,n}$ of $P_n(z)$ are indexed relative to the center $-1/2$. As $n \to \infty$, the zeros follow a "clock" pattern with spacing of order $O(1/n)$.
  • For even $n$: $z_{j,2n} = -\frac{1}{2} + \frac{\pi^2}{4} \frac{j}{n+1} + o(\frac{1}{n})$.
  • For odd $n$: $z_{j,2n+1} = -\frac{1}{2} + \frac{\pi^2}{8} \frac{2j-1}{n+1} + o(\frac{1}{n})$.
• Conclusion: The distance between consecutive zeros in the bulk is asymptotically $\frac{\pi^2}{4n}$.
• Theorem 8: Analogous results are provided for Eulerian polynomials, showing their zeros in $(-\infty, 0)$ are distributed with a spacing of $\frac{\pi^2}{n}$.

D. Orthogonality Properties

The paper proves new orthogonality relations based on the symmetry of the polynomials.

• Lemma 2 & 5: The authors demonstrate that Geometric and Eulerian polynomials satisfy specific orthogonality conditions with respect to weight functions involving powers of $(1+z)$ or $(1-x)$. For instance, integrals of products of polynomials with different parities over $(-1, 0)$ vanish.

4. Significance

• Unification of Combinatorial Objects: The paper rigorously connects the asymptotic analysis of Geometric polynomials (related to set partitions) and Eulerian polynomials (related to permutations), showing that their zero distributions share fundamental structural similarities.
• Precision in Zero Distribution: While the existence of real zeros for these polynomials was known, this paper provides the exact asymptotic spacing and the error terms ($o(1/n)$), offering a "clock behavior" description similar to that found in orthogonal polynomials (like Legendre or Chebyshev).
• Analytic Framework: The derivation of the asymptotic formula involving the function $J_n(z)$ provides a powerful tool for approximating these polynomials in the complex plane, which is useful for numerical analysis and further combinatorial studies.
• Extension to Generalized Forms: The authors note that these results can be extended to polynomials defined by linear fractional transformations of $P_n(z)$, suggesting a broader applicability of their methods.

In summary, the paper provides a comprehensive analytic framework for understanding the large-$n$ behavior of Geometric and Eulerian polynomials, moving from global complex asymptotics to the local, precise distribution of their zeros on the real line.