相关论文: Congruences on Stirling numbers and Eulerian numbe…
Extensions of the $Stirling$ numbers of the second kind and $Dobinski$ -like formulas are proposed in a series of exercises for graduates. Some of these new formulas recently discovered by me are to be found in the source paper $ [1]$.…
In this paper, we investigate the spherically symmetric Finsler metrics with isotropic S-curvature and obtain a characterized equation. As an application, we prove that these metrics with Douglas type must be Randers metrics or Berwald…
We give congruences between the Eisenstein series and a cusp form in the cases of Siegel modular forms and Hermitian modular forms. We should emphasize that there is a relation between the existence of a prime dividing the $k-1$-th…
We prove three variations of recent results due to Andrews on congruences for $NT(m,k,n)$, the total number of parts in the partitions of $n$ with rank congruent to $m$ modulo $k$. We also conjecture new congruences and relations for…
The Stirling numbers of the first kind can be represented in terms of a new class of polynomials that are closely related to the Bernoulli polynomials. Recursion relations for these polynomials are given.
In this present paper, we show that the Stirling numbers of the first kind with higher level connected with the probability distribution of the number of records and record times in the so-called F^{\alpha}-scheme. In addition, we determine…
Let P be the set of the sequence of polynomials of degree n. The aim of this paper is to study the Stirling numbers of the second kind associated with P and of the first kind associated with P, in a unified and systematic way with the help…
We introduce a generalization of the Stirling numbers via symmetric functions involving two weight functions. The resulting extension unifies previously known Stirling-type sequences with known symmetric function forms, as well as other…
By means of the generating function method, a linear recurrence relation is explicitly resolved. The solution is expressed in terms of the Stirling numbers of both the first and the second kind. Two remarkable pairs of combinatorial…
Spivey's combinatorial method revealed an important identity for Bell numbers, involving Stirling numbers of the second kind. This paper extends his work by deriving Spivey-type recurrence relations for fully degenerate Bell polynomials and…
In these notes I proved the Chai-Faltings version of Eichler-Shimura congruence relation for simple GSpin Shimura varieties. This extends the results by Bueltel, Wedhorn and Koskivirta.
In the paper, the author finds an explicit formula for computing Bell numbers in terms of Kummer confluent hypergeometric functions and Stirling numbers of the second kind.
In this paper we prove some transformation formulae for congruences modulo a prime and deduce some congruences for Domb numbers and Almkvist-Zudilin numbers. We also pose some conjectures on congruences modulo prime powers.
Based on a $q$-congruence of the author and Petrov, we set up a $q$-analogue of Sun--Tauraso's congruence for sums of Catalan numbers, which extends a $q$-congruence due to Tauraso.
In this paper, we derive some interesting symmetric properties for the geenralized Euler numbers and polynomials.
The second Eulerian numbers are defined via the descent enumerator of Stirling permutations, a class of permutations introduced by Gessel and Stanley. We give a simple and conceptual proof of two identities relating the Bernoulli numbers…
In the paper, the authors review some explicit formulas and establish a new explicit formula for Bernoulli and Genocchi numbers in terms of Stirling numbers of the second kind.
Recently Andrews and Sellers proved some amazing congruences for the Fishburn numbers. We extend their results to a more general sequence of numbers. As a result we prove a new congruence mod 23 for the Fishburn numbers and prove their…
In this paper we obtain some sophisticated combinatorial congruences involving binomial coefficients and confirm two conjectures of the author and Davis. They are closely related to our investigation of the periodicity of the sequence…
Recent developments of Baxter algebras have lead to applications to combinatorics, number theory and mathematical physics. We relate Baxter algebras to Stirling numbers of the first kind and the second kind, partitions and multinomial…