广义Stirling型数、Array型多项式、Eulerian型多项式的Generating functions及其应用

关于Dowling格的Whitney数的一些结果

Some results on Whitney numbers of Dowling lattices

本文研究了Dowling格的Whitney数及相关多项式的一些性质。我们回答了以下问题：斯特林多项式和欧拉多项式之间存在关系。我们能否找到道林多项式与推广欧拉多项式的其他多项式之间的新关系？此外，还给出了Dowling数的一些同余式.

Generalized Stirling Numbers I

We consider generalized Stirling numbers of the second kind $% S_{a,b,r}^{α_{s},β_{s},r_{s},p_{s}}\left( p,k\right) $, $% k=0,1,\ldots .rp+\sum_{s=2}^{L}r_{s}p_{s}$, where $a,b,α_{s},β_{s} $ are complex numbers, and $r,p,r_{s},p_{s}$ are non-negative integers given, $s=2,\ldots ,L$. (The case $a=1,b=0,r=1,r_{s}p_{s}=0$, corresponds to the standard Stirling numbers $S\left( p,k\right) $.) The numbers $% S_{a,b,r}^{α_{s},β_{s},r_{s},p_{s}}\left( p,k\right) $ are connected with a generalization of Eulerian numbers and polynomials we studied in previous works. This link allows us to propose (first…

An Aearated Triangular Array of Integers

Congruences modulo prime powers involving generalized Harmonic numbers are known. While looking for similar congruences, we have encountered a curious triangular array of numbers indexed with positive integers $n,k$, involving the Bernoulli and cycle Stirling numbers. These numbers are all integers and they vanish when $n-k$ is odd. This triangle has many similarities with the Stirling triangle. In particular, we show how it can be extended to negative indices and how this extension produces a {\it second kind} of such integers which may be considered as a new generalization of the Genocchi nu…

Riordan arrays and generalized Euler polynomials

Generalization of the Euler polynomials ${{A}_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{m}^{n}}{{x}^{m}}}$ are the polynomials ${{α}_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{u}_{n}}}\left( m \right){{x}^{m}}$, where ${{u}_{n}}\left( x \right)$ is the polynomial of degree $n$. These polynomials appear in various fields of mathematics, which causes a variety of methods for their study. In present paper we will consider generalized Euler polynomials as an attribute of the theory of Riordan arrays. From this point of vi…

满足某种超递推关系的三角阵列行的对数凹性

Log-concavity of rows of triangular arrays satisfying a certain super-recurrence

本文研究满足超递推关系的三角阵列，给出其行具有对数凹性的充分条件，统一证明多类经典组合序列的对数凹性并证实广义 Lah 数猜想，还引入双参数 Eulerian 数推广并讨论其性质。

Apostol-Dedekind 和相关和的指数生成函数

Exponential generating functions for Apostol-Dedekind and related sums

本文构造 Apostol-Dedekind 和的指数生成函数，定义高阶和及相关有限和族，并利用这些函数推导出许多新公式。

Computation Methods for combinatorial sums and Euler type numbers related to new families of numbers

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well-known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique in…

关于Lehmer提出的Comtet数问题

On a Question of Lehmer concerning the Comtet Numbers

本文将Comtet数视为加权斯特林多项式的特殊值，证明其满足四项递推式，回答了Lehmer关于$B(n,k)$是否存在固定项数递推的问题，并推导了Touchard型多项式的相关关系。

多重调和和 $\mathcal{H}_{\lbrace s\rbrace^{2l}=1;p-1}$ 模 $p^4$ 及其应用

Multiple harmonic sums $\mathcal{H}_{\lbrace s\rbrace^{2l}=1;p-1}$ modulo $p^4$ and applications

本文研究多重调和和模$p^4$，建立Wilson、Glaisher和Sun定理的推广，并分解Bernoulli数卷积的$p$-剩余，推广了Sun的结果。