Let {U n } be given by U 0 = 1 and U n = −2 [n/2] k=1 n 2k U n−2k (n ≥ 1), where [·] is the greatest integer function. Then {U n } is an analogy of Euler numbers and U 2n = 3 2n E 2n (1 3), where E m (x) is the Euler polynomial. In a previous paper the author gave many properties of {U n }. In the paper we present a summation formula and several congruences involving {U n }.

A Jesús Muñoz Díaz por su cumpleaños Contents 1. Introduction. 1 2. Elliptic sheaves and Hecke correspondences. Euler products 3 2.1. Elliptic sheaves 3 2.2. ∞-Level structures 5 2.3. Hecke correspondences 6 2.4. Euler products 8 3. Isogenies and Hecke correspondences 9 3.1. Isogenies for elliptic sheaves 9 3.2. Trivial correspondences 11 4. Ideal class group anhilators for the cyclotomic funct...

We are concerned with the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as the following second-order nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential φ : Ω ⊂ R → R: ...

Here presented is the interrelationship between Eulerian polynomials, Eulerian fractions and Euler-Frobenius polynomials, Euler-Frobenius fractions, Bsplines, respectively. The properties of Eulerian polynomials and Eulerian fractions and their applications in B-spline interpolation and evaluation of Riemann zeta function values at odd integers are given. The relation between Eulerian numbers a...

Abstract We recall two formulas, due to C. Jordan, for the successive derivatives of functions with an exponential or logarithmic inner function. We apply them to get addition formulas for the Stirling numbers of the second kind and for the Stirling numbers of the first kind. Then we show how one can obtain, in a simple way, explicit formulas for the generalized Euler polynomials, generalized E...

In this paper, we investigate those positive integers n for which the equality σ(φ(n)) = σ(n) holds, where σ is the sum of the divisors function and φ is the Euler function.

In this paper, by using the Euler-Maclaurin expansion for the zeta function and estimating the weight function effectively, we derive a strengthenment of a Hardy-Hilbert’s type inequality proved by W.Y. Zhong. As applications, some particular results are considered.

Let φ(·) be the Euler function and let σ(·) be the sum-of-divisors function. In this note, we bound the number of positive integers n x with the property that s(n) = σ(n)− n divides φ(n).