Related papers: An Euler-type formula for partitions of the M\"obi…

In this paper, we study various classes of partition functions such as those related to the parity of the number of parts, to differences of partition numbers, and to partitions with a repeated smallest part. We establish identities…

We review the well-known slice theorem of Ebin for the action of the diffeomorphism group on the space of Riemannian metrics of a closed manifold. We present advances in the study of the spaces of Riemannian metrics, and produce a more…

We prove that the Euler-Chow series for ruled surfaces and scrolls is rational by means of an explicit computation.

Euler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$, \begin{align*}…

We equip the space of lattice cones with a coproduct which makes it a connected cograded colagebra. The exponential sum and exponential integral on lattice cones can be viewed as linear maps on this space with values in the space of…

In this paper, we generalized De Moivre's formula and Euler's formula to octonions and find the roots of generalized octonions using these formulae.

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 give an Euler Maclaurin formula with remainder for the sum of the values of a smooth function on the integral points in a simple integral polytope. This formula is proved by elementary methods.

This note contains some asymptotic formulas for the sums of various residue classes of Euler's phi-function.

This paper explores the properties of multipliers associated with discrete analogues of fractional integrals, revealing intriguing connections with Dirichlet characters, Euler's identity, and Dedekind zeta functions of quadratic imaginary…

We show that $$ \sum_{n\neq m}\frac{\mu(n)\mu(m)}{nm}E_{X}\left(\{nx\}\{mx\}\right)=-\frac{9}{2\pi^{2}}+O\left(\frac{1}{X}\right), $$ where $x$ is uniformly distributed in $[0,X]$ with $X\in \mathbb{N}$, $E_{X}(.)$ denotes the expected…

The aim of this paper is twofold. One is to give a definition of the Euler characteristic of infinite acyclic categories with filtrations and the other is to prove the invariance of the Euler characteristic under the subdivision of finite…

We present a largely self contained account on the K-theory of a weighted smooth projective curve over an algebraically closed field. In particular, we discuss the weighted version of divisor theory, Euler form, and Riemann-Roch theorem.…

In this article we give a result obtained of an experimental way for the Euler totient function.

In the note, the author discovers an explicit formula for computing Bernoulli numbers in terms of Stirling numbers of the second kind.

Glaisher's theorem states that the number of partitions of $n$ into parts which repeat at most $m-1$ times is equal to the number of partitions of $n$ into parts which are not divisible by $m$. The $m=2$ case is Euler's famous partition…

An unusual formula for the Euler characteristics of even dimensional triangulated manifolds is deduced from the generalized Dehn-Sommerville equations.

In this paper, we give a conjecture, which generalises Euler's partition theorem involving odd parts and different parts for all moduli. We prove this conjecture for two family partitions. We give $q$-difference equations for the related…

In this paper a new approach is derived in the context of shape theory. The implemented methodology is motivated in an open problem proposed in \citet{GM93} about the construction of certain shape density involving Euler hypergeometric…

An equation containing a fractional power of an elliptic operator of second order is studied for Dirichlet boundary conditions. Finite difference approximations in space are employed. The proposed numerical algorithm is based on solving an…