The asymptotic properties of Eulerian numbers and refined Eulerian numbers: A Spline perspective

TL;DR

This paper derives improved asymptotic formulas for Eulerian and refined Eulerian numbers using spline interpretations, connecting these combinatorial numbers to B-splines and Hermite polynomials, and enhancing convergence accuracy.

Contribution

It introduces a spline-based approach to obtain asymptotic formulas for Eulerian numbers with better convergence and extends this to refined Eulerian numbers using Hermite polynomial representations.

Findings

Asymptotic formulas match previous results but with improved convergence.

Spline techniques effectively derive properties of Eulerian numbers.

Refined Eulerian numbers are represented asymptotically via Hermite polynomials.

Abstract

In this paper, the asymptotic formulas for Eulerian numbers, refined Eulerian numbers and the coefficients of descent polynomials are obtained directly from the spline interpretations of these numbers. Having related these numbers directly to B-splines [15], we can take advantage of many powerful spline techniques to derive various results of these numbers. The asymptotic formulas for the Eulerian numbers Ad;k agree with the previously known results which were given by L. Carlitz et al.(1972)[2] and S.Tanny (1973) [18], but the convergence order is much better. We also give the asymptotic representations of refined Eulerian numbers which is in terms of the Hermite polynomials.