Related papers: Shifted Appell sequences in Clifford analysis

Let $P_1,\dots,P_k \colon {\bf Z} \to {\bf Z}$ be polynomials of degree at most $d$ for some $d \geq 1$, with the degree $d$ coefficients all distinct, and admissible in the sense that for every prime $p$, there exists integers $n,m$ such…

Very recently, Tu et al. presented a sufficient condition about $(a_1,a_2,a_3)$, see Theorem 1.1, such that $f(x) = x^{3\cdot 2^m} + a_1 x^{2^{m+1}+1}+ a_2 x^{2^m+2} + a_3 x^3$ is a class of permutation polynomials over $\gf_{2^{n}}$ with…

We study the uniform distribution of the polynomial sequence $\lambda(P)=(\lfloor P(k) \rfloor )_{k\geq 1}$ modulo integers, where $P(x)$ is a polynomial with real coefficients. In the nonlinear case, we show that $\lambda(P)$ is uniformly…

We consider a polynomial $P\in \mathbb{R}[x_{1},\cdots, x_{d}]$ of degree $ \delta $ that depends non-trivially on each of $x_1,...,x_d$ with $d\geq 2$. For any integer $t$ with $2\leq t\leq d$, any natural number $n \in \mathbb{N}$, and…

Suppose that $f(x)=x^4+Ax^3+Bx^2+Ax+1\in {\mathbb Z}[x]$. We say that $f(x)$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\theta^3\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where…

We extend constructions of classical Clifford analysis to the case of indefinite non-degenerate quadratic forms. Clifford analogues of complex holomorphic functions - called monogenic functions - are defined by means of the Dirac operators…

In this research, as the new results of our previously proposed definition for the new class of $2D$ $q$-Appell polynomials, we derive some interesting relations including the recurrence relation and partial $q$-difference equation of the…

Several versions of the Fourier transform have been formulated in the framework of Clifford algebra. We present a (Clifford-Fourier) transform, constructed using the geometric properties of Clifford algebra. We show the corresponding…

Let $\mathbb{Z}_K$ denote the ring of integers of the number field $K = \mathbb{Q}(\theta)$, where $\theta$ is a root of the monic irreducible polynomial $f(x) \in \mathbb{Z}[x]$. We say that $f(x)$ is monogenic if $\mathbb{Z}_K =…

Quaternionic Clifford analysis is a recent new branch of Clifford analysis, a higher dimensional function theory which refines harmonic analysis and generalizes to higher dimension the theory of holomorphic functions in the complex plane.…

We are interested in finding a nonlinear polynomial $P$ on $\mathbb{R}^n$ that solves the minimal surface equation. Even though no explicit solution is found in this article, we investigate constraints that a polynomial solution must obey.…

Let ${\mathcal P}_k$ denote the set of all algebraic polynomials of degree at most $k$ with real coefficients. Let ${\mathcal P}_{n,k}$ be the set of all algebraic polynomials of degree at most $n+k$ having exactly $n+1$ zeros at $0$. Let…

Let $K$ be a field of characteristic zero and suppose that $f:\mathbb{N}\to K$ satisfies a recurrence of the form $$f(n)\ =\ \sum_{i=1}^d P_i(n) f(n-i),$$ for $n$ sufficiently large, where $P_1(z),...,P_d(z)$ are polynomials in $K[z]$.…

In this paper we consider in detail the composition of an irreducible polynomial with X^2 and suggest a recurrent construction of irreducible polynomials of fixed degree over finite fields of odd characteristics. More precisely, given an…

We consider orthogonal polynomials with respect to a linear differential operator $$\mathcal{L}^{(M)}=\sum_{k=0}^{M}\rho_{k}(z)\frac{d^k}{dz^k}, $$ where $\{\rho_k\}_{k=0}^{M}$ are complex polynomials such that $deg[\rho_k]\leq k, 0\leq k…

The first author introduced a sequence of polynomials (\cite{8}, sequence A174531) defined recursively. One of the main results of this study is proof of the integrality of its coefficients.

In 2012, for any integer $n \ge 2$, Kedlaya constructed an infinite class of monic irreducible polynomials of degree $n$ with integer coefficients having squarefree discriminants. Such polynomials are necessarily monogenic. Further, by…

We show that any Appell sequence can be written in closed form as a forward difference transformation of the identity. Such transformations are actually multipliers in the abelian group of the Appell polynomials endowed with the operation…

While considering nonlinear coherent states with specific anti-holomorphic coefficients $\bar{z}^n/\sqrt{x_n!}$, we identify as first associated Meixner-Pollaczek polynomials the orthogonal polynomials arising from shift operators which are…

In this article, we propose a few sufficient conditions on polynomials having integer coefficients all of whose zeros lie outside a closed disc centered at the origin in the complex plane and deduce the irreducibility over the ring of…