Congruences on Stirling numbers and Eulerian numbers

Not so new congruences for Stirling numbers of the first kind, with an application to Chern classes

In this paper we give simple expressions, involving binomials coefficients, for the value of $c(n,k)$ modulo $p^{v_p(n)}$, when $v_p(n) > 0$. Here $c(n,k)$ denotes a Stirling number of the first kind, and $v_p(n)$ is the highest power of $p$ dividing $n$. As an application, we compute the Chern classes of permutation representations of cyclic groups.

Congruences related to Eulerian and Euler numbers

Using Eulerian and Euler numbers, we establish congruences concerning sums involving harmonic numbers, tangent numbers and Genocchi numbers.

Stirling numbers with level $2$ and poly-Bernoulli numbers with level $2$

In this paper, we introduce poly-Bernoulli numbers with level $2$, related to the Stirling numbers of the second kind with level $2$, and study several properties of poly-Bernoulli numbers with level $2$ from their expressions, relations, and congruences. Poly-Bernoulli numbers with level $2$ have strong connections with poly-Cauchy numbers with level $2$. In a special case, we can determine the denominators of Bernoulli numbers with level $2$ by showing a von Staudt-Clausen like theorem.

Stirling's approximation and a hidden link between two of Ramanujan's approximations

A conjectured relation between Ramanujan's asymptotic approximations to the exponential function and the exponential integral is established. The proof involves Stirling numbers, second-order Eulerian numbers, modifications of both of these, and Stirling's approximation to the gamma function. Our work provides new information about the coefficients in Stirling's approximation and their connection to Ramanujan's approximation coefficients. A more analytic second proof of the main result is also included in an appendix.

Some results on Whitney numbers of Dowling lattices

In this paper, we study some properties of Whitney numbers of Dowling lattices and related polynomials. We answer the following question: there is relation between Stirling and Eulerian polynomials. Can we find a new relation between Dowling polynomials and other polynomials generalizing Eulerian polynomials? In addition, some congruences for the Dowling numbers are given.

Generalized degenerate Bernoulli numbers and polynomials arising from Gauss hypergeometric function

In a previous paper, Rahmani introduced a new family of p-Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. Motivated by this paper and as a degenerate version of those numbers and polynomials, we introduce the generalized degenerate Bernoulli numbers and polynomials again by using the Gauss hypergeometric function. In addition, we introduce the degenerate type Eulerian numbers as a degenerate version of Eulerian numbers. For the generalized degenerate Bernoulli numbers, we express them in terms of the degenerate Stirling numbers of the second kind, of the degene…

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…

混合型贝尔数的素数与 Touchard 同余式

Prime and Touchard Congruences of Mixed-Type Bell Numbers

本文建立混合型斯特林数与贝尔数的组合算术框架，连通分拆结构、Touchard 多项式与素幂同余式，结合 p 进赋值理论证明相关同余式，推广组合序列经典算术性质。

Divisibility by 2 of partial Stirling numbers

The partial Stirling numbers T_n(k) used here are defined as the sum over odd values of i of (n choose i) i^k. Their 2-exponents nu(T_n(k)) are important in algebraic topology. We provide many specific results, applying to all values of n, stating that, for all k in a certain congruence class mod 2^t, nu(T_n(k)) = nu(k - k0) + c0, where k0 is a 2-adic integer and c0 a positive integer. Our analysis involves several new general results for nu(sum (n choose 2i+1) i^j), the proofs of which involve a new family of polynomials. Following Clarke, we interpret T_n as a function on the 2-adic integers…

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…

第二类混合斯特林数的算术性质

Arithmetic Properties of Mixed Stirling Numbers of the second kind

本文研究第二类混合斯特林数，建立其递推关系与指数生成函数，分析其模素数$p$及$p^2$的性质，推广经典Touchard同余式，揭示其具有区别于经典集合分拆的独特数论特征。

退化 Eulerian 多项式与数

Degenerate Eulerian polynomials and numbers

本文引入并研究退化 Eulerian 多项式与退化 Eulerian 数，推导其恒等式、递推关系、生成函数及退化 Worpitzky 恒等式，并建立其与退化 Stirling 数和 Bernoulli 数的联系。

The $p$-adic Analysis of Stirling Numbers via Higher Order Bernoulli Numbers

In this paper, we use our previous study of the higher order Bernoulli numbers $B_n^{(l)}$ to investigate the $p$-adic properties of the Stirling numbers of the second kind $S(n,k)$. For example, we give a new, greatly simplified proof of the formula $ν_2(S(2^h,k))=d_2(k)-1$ if $1\le k \le 2^h$, and generalize this result to arbitrary primes $p$. We also consider the Stirling numbers of the first kind $s(n,k)$, with new results analogous to those for the Stirling numbers of the second kind. New mod $p$ congruences for Stirling numbers of both kinds are also given.

On the $p$-adic properties of Stirling numbers of the first kind

Let $n, k$ and $a$ be positive integers. The Stirling numbers of the first kind, denoted by $s(n,k)$, count the number of permutations of $n$ elements with $k$ disjoint cycles. Let $p$ be a prime. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, Adelberg, Hong and Qiu made some progress in the study of the $p$-adic valuations of $s(n,k)$. In this paper, by using Washington's congruence on the generalized harmonic number and the $n$-th Bernoulli number $B_n$ and the properties of $m$-th Stirling numbers of the first kind obtained recently by the authors, we arrive at an exact ex…