B样条在一些渐近组合问题中的应用

The applications of B-splines in some asymptotic combinatorial problems

本文考察了B样条函数及其导数的渐近性质,并给出了收敛阶;考察了经典Eulerian数和两类广义Eulerian数的渐近性质;给出了以Hermite多项式表示的细化Eulerian数的渐近形式.Carlitz等人利用中心极限定理得到Eulerian数渐近公式的逼近阶为43阶.利用样条方法,我们得到更为精确的逼近阶.将样条方法引入到组合数的渐近分析中,为离散对象的研究提供了一种新的分析方法.

In this paper, the convergence order of the derivatives of B-splines is investigated. 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. In terms of Hermite polynomials, the asymptotic representations of refined Eulerian numbers are also concluded. The asymptotic formulas for the Eulerian numbers Ad,k agree with the previously known results which were given by Carlitz et al., but the convergence order is more accurate. This paper also provides applications of B-splines in the asymptotic combinatorics.