Related papers: $*$-exponential of slice-regular functions

We study automorphisms and invariants for the algebra $\mathbb{O}$ of octonions and octonionic slice regular functions $f:\mathbb{O} \to \mathbb{O}$.

We study explicit continued fraction expansions for certain series. Some of these expansions have symmetry that generalizes some remarkable examples discovered independently by Kmosek and Shallit. Furthermore, we prove the following…

Let $f$ be a transcendental entire function. The quite fast escaping set, $Q(f)$, and the set $Q_2(f),$ which was defined recently, are equal to the fast escaping set, $A(f),$ under certain conditions. In this paper we generalise these sets…

The present paper proves that if for a power sum $\alpha$ over $\ZZ$ the length of the period of the continued fraction for $\sqrt{\alpha(n)}$ is constant for infinitely many even (resp. odd) $n$, then $\sqrt{\alpha(n)}$ admits a functional…

We prove the following superexponential distribution inequality: for any integrable $g$ on $[0,1)^{d}$ with zero average, and any $\lambda>0$ \[ |\{ x \in [0,1)^{d} \; :\; g \geq\lambda \}| \leq e^{-…

We present some new relations between the Cauchy-Riemann operator on the real Clifford algebra $\mathbb R_n$ of signature $(0,n)$ and slice-regular functions on $\mathbb R_n$. The class of slice-regular functions, which comprises all…

The Fourier transform is naturally defined for integrable functrions. Otherwise, it should be stipulated in which sense the Fourier transform is understood. We consider some class of radial and, generally saying, nonintegrable functions.…

Let $g$ be a entire function of exponential type on the complex plane $\mathbb C$, $Z=\{ z_k\}_{k=1,2,\dots}$ be a sequence of points in $\mathbb C$. We give a criterion for the existence of an entire function $f\neq 0$ of exponential type…

A function f is continuous iff the PRE-image f^{-1}[V] of any open set V is open again. Dual to this topological property, f is called OPEN iff the IMAGE f[U] of any open set U is open again. Several classical Open Mapping Theorems in…

In this paper we characterize the closed invariant subspaces for the ($*$-)multiplier operator of the quaternionic space of slice $L^2$ functions. As a consequence, we obtain the inner-outer factorization theorem for the quaternionic Hardy…

Let g be a (say, sufficiently differentiable) function on the reals. One knows how to apply g to Hermitian elements A of a C* algebra. Yet the question of differentiability of the mapping A to g(A) is not trivial, since in general "A and dA…

In Combinatorics Stirling numbers may be defined in several ways. One such definition is given in [1], where an extensive consideration of Stirling numbers is presented. In this paper an alternative definition of Stirling numbers of both…

The recent generalizations of Boltzmann-Gibbs statistics mathematically relies on the deformed logarithmic and exponential functions defined through some deformation parameters. In the present work, we investigate whether a deformed…

Using model theory and differential algebra, we give necessary conditions for algebraic ordinary differential equations to have a complex Pfaffian solution on some complex domain. These tools also allow us to give many examples of algebraic…

An iterative square root of a function $f$ is a function $g$ such that $g(g(\cdot))=f(\cdot)$. We obtain new characterizations for detecting the non-existence of such square roots for self-maps on arbitrary sets. This is used to prove that…

This short note provides an explicit description of the Fr\'echet derivatives of the principal square root matrix functional at any order. We present an original formulation that allows to compute sequentially the Fr\'echet derivatives of…

In this note functions that transform open segments of a linear space into open segments of another linear space are studied and characterized. Assuming that the range is non-collinear, it is proved that such a map can always be expressed…

We give a formula for $f(\eta)$, where $f :\mathbb C \to \mathbb C$ is a continuously differentiable function satisfying $f(\bar z) = \overline{f(z)}$, and $\eta$ is a dual quaternion. Note this formula is straightforward or well known if…

Most of theoretical physics is based on the mathematics of functions of a real or a complex variable; yet we frequently are drawn to try extending our reach to include quaternions. The non-commutativity of the quaternion algebra poses…

We present a theorem on taking the repeated indefinite summation of a holomorphic function $\phi(z)$ in a vertical strip of $\mathbb{C}$ satisfying exponential bounds as the imaginary part grows. We arrive at this result using transforms…