scholarly journals Some Identities Involving Second Kind Stirling Numbers of Types $B$ and $D$

10.37236/8703 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Eli Bagno ◽  
Riccardo Biagioli ◽  
David Garber
Keyword(s):  
Polynomial Ring ◽  
Transition Matrix ◽  
Stirling Numbers ◽  
Eulerian Numbers ◽  
Standard Bases ◽  
Definition Of ◽  
Colored Permutations

Using Reiner's definition of Stirling numbers of the second kind in types $B$ and $D$, we generalize two well-known identities concerning the classical Stirling numbers of the second kind. The first identity relates them with Eulerian numbers and the second identity interprets them as entries in a transition matrix between the elements of two standard bases of the polynomial ring $\mathbb{R}[x]$. Finally, we generalize these identities to the group of colored permutations $G_{m,n}$.

EIGENTIME IDENTITY OF THE WEIGHTED KOCH NETWORKS

Fractals ◽  
2018 ◽  
Vol 26 (03) ◽  
pp. 1850042 ◽  
Author(s):  
YU SUN ◽  
JIAHUI ZOU ◽  
MEIFENG DAI ◽  
XIAOQIAN WANG ◽  
HUALONG TANG ◽  
...  
Keyword(s):  
Transition Matrix ◽  
Small World ◽  
Laplacian Matrix ◽  
Block Matrix ◽  
The Other ◽  
Small World Networks ◽  
Eigenvalues And Eigenvectors ◽  
Two Factors ◽  
Analytical Research ◽  
Definition Of

The eigenvalues of the transition matrix of a weighted network provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to biased walks. Although various dynamical processes have been investigated in weighted networks, analytical research about eigentime identity on such networks is much less. In this paper, we study analytically the scaling of eigentime identity for weight-dependent walk on small-world networks. Firstly, we map the classical Koch fractal to a network, called Koch network. According to the proposed mapping, we present an iterative algorithm for generating the weighted Koch network. Then, we study the eigenvalues for the transition matrix of the weighted Koch networks for weight-dependent walk. We derive explicit expressions for all eigenvalues and their multiplicities. Afterwards, we apply the obtained eigenvalues to determine the eigentime identity, i.e. the sum of reciprocals of each nonzero eigenvalues of normalized Laplacian matrix for the weighted Koch networks. The highlights of this paper are computational methods as follows. Firstly, we obtain two factors from factorization of the characteristic equation of symmetric transition matrix by means of the operation of the block matrix. From the first factor, we can see that the symmetric transition matrix has at least [Formula: see text] eigenvalues of [Formula: see text]. Then we use the definition of eigenvalues and eigenvectors to calculate the other eigenvalues.

Ideals Modulo a Prime

2020 ◽  
pp. 2150042
Author(s):  
John Abbott ◽  
Anna Maria Bigatti ◽  
Lorenzo Robbiano
Keyword(s):  
Prime Number ◽  
Polynomial Ring ◽  
Distinct Advantage ◽  
Ideal Formula ◽  
Reduction Modulo ◽  
Good For ◽  
Definition Of ◽  
The Given ◽  
Term Ordering

The main focus of this paper is on the problem of relating an ideal [Formula: see text] in the polynomial ring [Formula: see text] to a corresponding ideal in [Formula: see text] where [Formula: see text] is a prime number; in other words, the reduction modulo[Formula: see text] of [Formula: see text]. We first define a new notion of [Formula: see text]-good prime for [Formula: see text] which does depends on the term ordering [Formula: see text], but not on the given generators of [Formula: see text]. We relate our notion of [Formula: see text]-good primes to some other similar notions already in the literature. Then we introduce and describe a new invariant called the universal denominator which frees our definition of reduction modulo [Formula: see text] from the term ordering, thus letting us show that all but finitely many primes are good for [Formula: see text]. One characteristic of our approach is that it enables us to easily detect some bad primes, a distinct advantage when using modular methods.

scholarly journals Construction of the Type 2 Poly-Frobenius-Genocchi Polynomials with Their Certain Applications

2020 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Serkan Araci
Keyword(s):  
Linear Combination ◽  
Generating Function ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Genocchi Polynomials ◽  
Definition Of ◽  
Special Case

Motivated by the definition of the type 2 poly-Bernoulli polynomials introduced by Kim-Kim, in the present paper, we consider a class of new generating function for the Frobenius-Genocchi polynomials, called the type 2 poly-Frobenius-Genocchi polynomials, by means of the polyexponential function. Then, we derive some useful relations and properties. We show that the type 2 poly-Frobenius-Genocchi polynomias equal a linear combination of the classical Frobenius-Genocchi polynomials and Stirling numbers of the first kind. In a special case, we give a relation between the type 2 poly-Frobenius-Genocchi polynomials and Bernoulli polynomials of order k. Moreover, inspired by the definition of the unipoly-Bernoulli polynomials introduced by Kim-Kim, we introduce the unipoly-Frobenius-Genocchi polynomials by means of unipoly function and give multifarious properties including derivative and integral properties. Furthermore, we provide a correlation between the unipoly-Frobenius-Genocchi polynomials and the classical Frobenius-Genocchi polynomials.

RESTRICTED RANDOM WALK NUMBERS OF THE FIRST AND SECOND KIND

2004 ◽  
Vol 03 (02) ◽  
pp. 155-162 ◽  
Author(s):  
A. S. PADMANABHAN
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Stirling Numbers ◽  
Exact Enumeration ◽  
Finite Memory ◽  
Definition Of

In a manner analogous to the definition of Stirling numbers of the first and second kind we define a set of numbers for restricted random walks of finite memory. These two sequences of counting numbers should be useful in the exact enumeration of properties of simple and restricted random walks.

scholarly journals Generalized degenerate Bernoulli numbers and polynomials arising from Gauss hypergeometric function

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Lee-Chae Jang ◽  
Hyunseok Lee ◽  
Hanyoung Kim
Keyword(s):  
Hypergeometric Function ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Gauss Hypergeometric Function ◽  
Eulerian Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
New Family ◽  
Degenerate Type ◽  
Degenerate Version

AbstractA new family of p-Bernoulli numbers and polynomials was introduced by Rahmani (J. Number Theory 157:350–366, 2015) with the help of the Gauss hypergeometric function. Motivated by that paper and in the light of the recent interests in finding degenerate versions, we construct the generalized degenerate Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. In addition, we construct 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 degenerate type Eulerian numbers, of the degenerate p-Stirling numbers of the second kind and of an integral on the unit interval. As to the generalized degenerate Bernoulli polynomials, we represent them in terms of the degenerate Stirling polynomials of the second kind.

Stirling Numbers and Eulerian Numbers

2016 ◽  
pp. 105-151 ◽  
Author(s):  
Carlo Mariconda ◽  
Alberto Tonolo
Keyword(s):  
Stirling Numbers ◽  
Eulerian Numbers

scholarly journals Congruences for Stirling numbers and Eulerian numbers

2008 ◽  
Vol 132 (4) ◽  
pp. 315-328
Author(s):  
Hui-Qin Cao ◽  
Hao Pan
Keyword(s):  
Stirling Numbers ◽  
Eulerian Numbers

scholarly journals A Novel Kind of the Type 2 Poly-Fubini Polynomials and Numbers

2020 ◽  
Author(s):  
Waseem Khan
Keyword(s):  
Linear Combination ◽  
Generating Function ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
New Class ◽  
Definition Of ◽  
Special Case

Motivation by the definition of the type 2 poly-Bernoulli polynomials introduced by Kim-Kim [9], in the present paper, we consider a new class of new generating function for the Fubini polynomials, called the type 2 poly-Fubini polynomials by means of the polyexponential function. Then, we derive some useful relations and properties. We show that the type 2 poly-Fubini polynomials equal a linear combination of the classical of the Fubini polynomials and Stirling numbers of the first kind. In a special case, we give a relation between the type 2 poly-Fubini polynomials and Bernoulli polynomials of order r. Moreover, inspired by the definition of the unipoly-Bernoulli polynomials introduced by Kim-Kim [9]. We introduce the type 2 unipoly-Fubini polynomials by means of unipoly function and give multifarious properties including derivative and integral properties. Furthermore, we provide a correlation between the unipoly-Fubini polynomials and the classical Fubini polynomials.

scholarly journals Flag Statistics from the Ehrhart $h^∗$-Polynomial of Multi-Hypersimplices

10.37236/5538 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Guo-Niu Han ◽  
Matthieu Josuat-Vergès
Keyword(s):  
Combinatorial Identities ◽  
Eulerian Numbers ◽  
Ehrhart Series ◽  
Unit Hypercube ◽  
Colored Permutations

It is known that the normalized volume of standard hypersimplices (defined as some slices of the unit hypercube) are the Eulerian numbers. More generally, a recent conjecture of Stanley relates the Ehrhart series of hypersimplices with descents and excedences in permutations. This conjecture was proved by Nan Li, who also gave a generalization to colored permutations. In this article, we give another generalization to colored permutations, using the flag statistics introduced by Foata and Han. We obtain in particular a new proof of Stanley’s conjecture, and some combinatorial identities relating pairs of Eulerian statistics on colored permutations.

scholarly journals A Research on a Certain Family of Numbers and Polynomials Related to Stirling Numbers, Central Factorial Numbers, and Euler Numbers

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
J. Y. Kang ◽  
C. S. Ryoo
Keyword(s):  
Complex Plane ◽  
Stirling Numbers ◽  
Euler Numbers ◽  
Interesting Phenomenon ◽  
Genocchi Numbers ◽  
Central Factorial Numbers ◽  
Definition Of

Recently, many mathematicians have studied different kinds of the Euler, Bernoulli, and Genocchi numbers and polynomials. In this paper, we give another definition of polynomialsŨn(x). We observe an interesting phenomenon of “scattering” of the zeros of the polynomialsŨn(x)in complex plane. We find out some identities and properties related to polynomialsŨn(x). Finally, we also derive interesting relations between polynomialsŨn(x), Stirling numbers, central factorial numbers, and Euler numbers.

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