Related papers: Functional calculus for dual quaternions
Using a summation identity obtained for the Fourier coefficients of $x^{2k}$, we derive a closed form expression for the zeta function at even positive integers, using a technique similar to one in an existing proof by Aladdi and Defant[1],…
We define a divisorial motivic zeta function for stable curves with marked points which agrees with Kapranov's motivic zeta function when the curve is smooth and unmarked. We show that this zeta function is rational, and give a formula in…
We represent the Euler alternating series (sometimes called the "Dirichlet eta function"), and generally $(b^s-b)\zeta(s)/b^s$ for $b>1$ an integer, in the half-plane $\Re s>0$, via series dominated by geometric series, with arbitrarily…
Derivations are linear operators which satisfy the Leibniz rule, while integrations are linear operators which satisfy the Rota-Baxter rule. In this paper, we introduce the notion of an FTC-pair, which consists of an algebra and module with…
The main outcomes of the paper are divided into two parts. First, we present a new dual for quadratic programs, in which, the dual variables are affine functions, and we prove strong duality. Since the new dual is intractable, we consider a…
A discrete complexified quaternion Fourier transform is introduced. This is a generalization of the discrete quaternion Fourier transform to the case where either or both of the signal/image and the transform kernel are complex…
In this paper we develop a functional calculus for a countable system of generators of contraction strongly continuous semigroups. As a symbol class of such calculus we use the algebra of polynomial tempered distributions. We prove a…
We define the zeta function of a finite category. And we propose a conjecture which states the relationship between the Euler characteristic of finite categories and the zeta function of finite categories. This conjecture is verified when…
We study the Felderhof free-fermion six-vertex model, whose wavefunction recently turned out to possess rich combinatorial structure of the Schur polynomials. We investigate the dual version of the wavefunction in this paper, which seems to…
This paper enriches preexisting satisfiability tests for unquantified languages, which in turn augment a fragment of Tarski's elementary algebra with unary real functions possessing a continuous first derivative. Two sorts of individual…
In this paper, we mainly establish the uncertainty principle (UP) for a function and its quaternion Fractional Fourier transform (QFrFT), as well as the UP for two QFrFTs. Using the polar representation of quaternion-valued signals, we give…
Motivated by the integral representation of the Euler Beta function, we introduce its Cauchy siblings and investigate some of their properties. Two of these newly introduced functions happen to coincide with some classical means, such as…
In this paper we study that the $q$-Euler numbers and polynomials are analytically continued to $E_q(s)$. A new formula for the Euler's $q$-Zeta function $\zeta_{E,q}(s)$ in terms of nested series of $\zeta_{E,q}(n)$ is derived. Finally we…
We give a simple formula for some determinants, and an analogous formula for pfaffians, both of which are polynomial identities. The second involve some expressions that interpolate between determinants and pfaffians. We give several…
The Kauffman Constraint Coefficients Kw and their corresponding Elementals Ew are presented as solutions to the construction of the (beta)-derivative of Kauffman's Theta-function. Additionally, a new recursion relation is provided to…
The so-called polynomial equations play an important role both in algebra and in the theory of functional equations. If the unknown functions in the equation are additive, relatively many results are known. However, even in this case, there…
In this paper, a derivative for functions $f : G \to H$, where $G$ is any metric divisible group and $H$ is a metric Abelian group with a group metric, is defined. Basic differentiation theorems are stated and demonstrated. In particular,…
We give an elementary combinatorial proof of Bass's determinant formula for the zeta function of a finite regular graph. This is done by expressing the number of non-backtracking cycles of a given length in terms of Chebychev polynomials in…
Let $f$ be a continuous real function defined in a subset of the real line. The standard definition of continuity at a point $x$ allow us to correlate any given epsilon with a (possibly depending of $x$) delta value. This pairing is known…
In this paper, we present a further generalization of the bi- periodic Fibonacci quaternions and octonions. We give the generating function, the Binet formula, and some basic properties of these quaternions and octonions. The results of…