Related papers: The Eulerian numbers can D.I.E
Dorpalen-Barry et al. proved Elser's conjecture about sign of Elser's number by interpreting them as certain sums of reduced Euler characteristics of an abstract simplicial complex known as $U$-nucleus complex. We prove a conjecture posed…
A classical result states that the parity balance of the number of excedances of all permutations (derangements, respectively) of length $n$ is the Euler number. In 2010, Josuat-Verg\`{e}s gives a $q$-analogue with $q$ representing the…
In the present paper, we introduce Eulerian polynomials attached to by using p-adic q-integral on Zp . Also, we give new interesting identities via the generating functions of Dirichlet's type of Eulerian polynomials. After, by applying…
We enumerate total cyclic orders on $\left\{1,\ldots,n\right\}$ where we prescribe the relative cyclic order of consecutive triples $(i,{i+1},{i+2})$, these integers being taken modulo $n$. In some cases, the problem reduces to the…
An interplay between the Lambert series and Euler's Pentagonal Number Theorem gives an Euler-type recurrence relation for any given arithmetical function. As consequences of this, we present Euler-type recurrence relations for some…
In this paper we study Eulerian extensions with edge constraints and use the probabilistic method to establish sufficient conditions for a given connected graph to be a subgraph of a Eulerian graph containing $m$ edges, for a given number…
The constant $\pi$ has fascinated scholars throughout the centuries, inspiring numerous formulas for its evaluation, such as infinite sums and continued fractions. Despite their individual significance, many of the underlying connections…
New series representations for odd powers of {\pi} i.e. {\pi}, {\pi}3 and {\pi}5 in terms of Euler numbers and even powers of {\pi} i.e. {\pi}2, {\pi}4 and {\pi}6 in terms of Bernoulli numbers have been obtained empirically.
Many kinds of convolution identities have been considered about several numbers, including Bernoulli, Euler, Genocchi, Cauchy, Stirling, and Fibonacci numbers. The well-known basic result about Bernoulli numbers is due to Euler. The…
We derive integral representations for six families of multiple Ap\'ery-like series using repeated integration by parts and Fourier expansions. The resulting formulas are expressed in terms of polylogarithms, Legendre chi functions, and…
We make connections of a counting problem of Eulerian cycles for undirected graphs to homological spectral graph theory, and formulate explicitly a trace formula that identifies the number of Eulerian circuits on an Eulerian graph with the…
3D stochastic Euler equations with a special form of multiplicative noise are considered. A Constantin-Iyer type representation in Euler-Lagrangian form is given, based on stochastic characteristics. Local existence and uniqueness of…
The numerical simulation of the 3D incompressible Euler equation is analyzed with respect to different integration methods. The numerical schemes we considered include spectral methods with different strategies for dealiasing and two…
We study several variants of Euler sums by using the methods of contour integration and residue theorem. These variants exhibit nice properties such as closed forms, reduction, etc., like classical Euler sums. In addition, we also define a…
The image of Euler's totient function is composed of the number 1 and even numbers. However, many even numbers are not in the image. We consider the problem of finding those even numbers which are in the image and those which are not. If an…
Translated from the Latin original, "Observationes generales circa series, quarum termini secundum sinus vel cosinus angulorum multiplorum progrediuntur" (1777). E655 in the Enestrom index. Euler looks at the binomial expansion $(1+x)^n$…
In this paper we consider a simple general form of a deterministic system with power-law memory whose state can be described by one variable and evolution by a generating function. A new value of the system's variable is a total (a…
Colored multiset Eulerian polynomials are a common generalization of MacMahon's multiset Eulerian polynomials and the colored Eulerian polynomials, both of which are known to satisfy well-studied distributional properties including…
In this study, we employ Euler's method and Richardson's extrapolation to solve a triple integral, which is then transformed into a third-order initial value problem. Our objective is to resolve the computational challenges associated with…
Consider a fluid flowing through a junction between two pipes with different sections. Its evolution is described by the 2D or 3D Euler equations, whose analytical theory is far from complete and whose numerical treatment may be rather…