This article presents a generalization of new classes degenerated Apostol–Bernoulli, Apostol–Euler, and Apostol–Genocchi Hermite polynomials level m. We establish some algebraic differential properties for generalizations Apostol–Bernoulli polynomials. These results are shown using generating function methods Apostol–Euler

Abstract The Riemann zeta distribution, defined as the one whose characteristic function is the normalised Riemann zeta function, is an interesting example of an infinitely divisible distribution. The infinite divisibility of the distribution has been proved with recourse to the Euler product of the Riemann zeta function. In this paper, we look at multiple zeta-star function, which is a multi-d...

In this paper: (i) We define and study a new numerical invariant R(X, g, ω) associated with a closed Riemannian manifold (M, g), a closed one form ω and a vector field X with isolated zeros. When X = − gradg f with f :M → R a Morse function this invariant is implicit in the work of Bismut–Zhang. The invariant is defined by an integral which might be divergent and requires (geometric) regulariza...

In this paper we study that the q-Euler numbers and polynomials are analytically continued to Eq(s). A new formula for the Euler’s q-Zeta function ζE,q(s) in terms of nested series of ζE,q(n) is derived. Finally we introduce the new concept of the dynamics of analytically continued q-Euler numbers and polynomials. 2000 Mathematics Subject Classification 11B68, 11S40 Key wordsq-Bernoulli polynom...

A generating function is given for the number, E(l, k), of irreducible kfold Euler sums, with all possible alternations of sign, and exponents summing to l. Its form is remarkably simple: ∑ n E(k + 2n, k) x n = ∑ d|k μ(d) (1 − x)/k, where μ is the Möbius function. Equivalently, the size of the search space in which k-fold Euler sums of level l are reducible to rational linear combinations of ir...

The beautiful Euler spiral, defined by the linear relationship between curvature and arclength, was first proposed as a problem of elasticity by James Bernoulli, then solved accurately by Leonhard Euler. Since then, it has been independently reinvented twice, first by Augustin Fresnel to compute diffraction of light through a slit, and again by Arthur Talbot to produce an ideal shape for a rail...

The double zeta function is a function of two arguments defined by a double Dirichlet series, and was first studied by Euler in response to a letter from Goldbach in 1742. By calculating many examples, Euler inferred a closed form evaluation of the double zeta function in terms of values of the Riemann zeta function, in the case when the two arguments are positive integers with opposite parity....

The basic properties of RSA cryptosystems and some classical attacks on them are described. Derived from geometric properties of the Euler functions, the Euler function rays, a new ansatz to attack RSA cryptosystems is presented. A resulting, albeit inefficient, algorithm is given. It essentially consists of a loop with starting value determined by the Euler function ray and with step width giv...