Fueter's theorem in discrete Clifford analysis

In this paper, we discretize techniques for the construction of axially monogenic functions to the setting of discrete Clifford analysis. Wherefore, we work in the discrete Hermitian Clifford setting, where each basis vector e_j is split into a forward and backward basis vector: . We prove a discrete version of Fueter's theorem in odd dimension by showing that for a discrete monogenic function f(ξ₀,ξ₁) left‐monogenic in two variables ξ₀ and ξ₁ and for a left‐monogenic P_k(ξ), the m‐dimensional function is in itself left monogenic, that is, a discrete function in the kernel of the discrete Dirac operator. Closely related, we consider a Vekua‐type system for the construction of axially monogenic functions. We consider some explicit examples: the discrete axial‐exponential functions and the discrete Clifford–Hermite polynomials. Copyright © 2015 John Wiley & Sons, Ltd.     

Hypercomplex derivative bases of polynomials in Clifford analysis

Convergence properties of hypercomplex derivative bases of special monogenic polynomials are studied. These new results extend and improve a lot of known works from the complex case to Clifford setting. Copyright © 2009 John Wiley & Sons, Ltd.     

The square root base of polynomials in Clifford analysis

In this paper the problem of taking the square root of bases of special monogenic polynomials is studied, thus leading to a number of results under some additional conditions of associated infinite matrices, related essentially to the so-called algebraicness of these matrices.     

Bernoulli polynomials and Pascal matrices in the context of Clifford analysis

This paper describes an approach to generalized Bernoulli polynomials in higher dimensions by using Clifford algebras. Due to the fact that the obtained Bernoulli polynomials are special hypercomplex holomorphic (monogenic) functions in the sense of Clifford Analysis, they have properties very similar to those of the classical polynomials. Hypercomplex Pascal and Bernoulli matrices are defined and studied, thereby generalizing results recently obtained by Zhang and Wang (Z. Zhang, J. Wang, Bernoulli matrix and its algebraic properties, Discrete Appl. Math. 154 (11) (2006) 1622-1632).     

Gel'fand–Tsetlin bases of orthogonal polynomials in Hermitean Clifford analysis

An explicit algorithmic construction is given for orthogonal bases for spaces of homogeneous polynomials, in the context of Hermitean Clifford analysis, which is a higher dimensional function theory centered around the simultaneous null solutions of two Hermitean conjugate complex Dirac operators. Copyright © 2011 John Wiley & Sons, Ltd.     

On solutions of a discretized heat equation in discrete Clifford analysis

The main purpose of this paper was to study solutions of the heat equation in the setting of discrete Clifford analysis. More precisely we consider this equation with discrete space and continuous time. Thereby we focus on representations of solutions by means of dual Taylor series expansions. Furthermore we develop a discrete convolution theory, apply it to the inhomogeneous heat equation and construct solutions for the related Cauchy problem by means of heat polynomials.     

On the Krall-type discrete polynomials

In this paper we present a unified theory for studying the so-called Krall-type discrete orthogonal polynomials. In particular, the three-term recurrence relation, lowering and raising operators as well as the second order linear difference equation that the sequences of monic orthogonal polynomials satisfy are established. Some relevant examples of q-Krall polynomials are considered in detail.     

The biaxial Fueter theorem

This paper deals with a special type of solutions for the Dirac operator on ? ^m , which can be obtained through a biaxial generalisation of the classical Fueter Theorem. This is a result which allows to generate zonal solutions for the Dirac equation starting from arbitrary holomorphic functions in the complex plane. Invoking operator identities for Jacobi polynomials, it is shown how this procedure can be extended to more general splittings than the one usually considered in the literature.     

On zeros of discrete orthogonal polynomials

We exploit difference equations to establish sharp inequalities on the extreme zeros of the classical discrete orthogonal polynomials, Charlier, Krawtchouk, Meixner and Hahn. We also provide lower bounds on the minimal distance between their consecutive zeros.     

Jacobi polynomials and generalized Clifford algebra‐valued Appell sequences

Appell sequences in Clifford analysis are defined as polynomial families on which the Heisenberg algebra acts through a raising and a lowering operator satisfying the canonical Heisenberg relation. Recently, these sequences have gained new interest, as they are connected to the topic of special functions (such as harmonic or monogenic Gegenbauer polynomials) and branching rules for certain irreducible representations of the spin group. In this paper, we will explain how Jacobi polynomials appear quite naturally in the setting of Appell sequences related to certain branching problems. Copyright © 2013 John Wiley & Sons, Ltd.     

Duality in complex Clifford analysis

The aim of this paper is to characterize the dual and bidual of complex Clifford modules of holomorphic functions which are defined over domains in $\text{C}$^{n + 1} and satisfy generalized Cauchy-Riemann equations. In one instance the generalized Cauchy-Riemann equation reduces to a holomorphic extension of Maxwell's equations in vacuo.     

Multiplicity of zeros and discrete orthogonal polynomials

We consider a problem of bounding the maximal possible multiplicity of a zero of some expansions Σ a_iF_i(x), at a certain point c, depending on the chosen family {imi}. The most important example is a polynomial with c = 1. It is shown that this question naturally leads to discrete orthogonal polynomials. Using this connection we derive some new bounds, in particular on the multiplicity of the zero at one of a polynomial with a prescribed norm. 30C15, 33C47     

Variational problems in Clifford analysis

Using decomposition results for Sobolev spaces of Clifford‐valued functions into direct sums of subspaces of monogenic and co‐monogenic functions variational problems will be studied.These variational problems are equivalent to PD‐models by the choice of special operators of conboundary differentiation. By a Galerkin scheme we construct the monogenic part as a weak solution of a non‐linear problem. The co‐monogenic potential is the solution of a weak Dirichlet problem. Copyright © 2002 John Wiley & Sons, Ltd.     

Bicomplex Hermitian Clifford analysis

Complex Hermitian Clifford analysis emerged recently as a refinement of the theory of several complex variables, while at the same time, the theory of bicomplex numbers motivated by the bicomplex version of quantum mechanics is also under full development. This stimulates us to combine the Hermitian Clifford analysis with the theory of bicomplex number so as to set up the theory of bicomplex Hermitian Clifford analysis. In parallel with the Euclidean Clifford analysis, the bicomplex Hermitian Clifford analysis is centered around the bicomplex Hermitian Dirac operator |D:C∞(R4n,W4n)→C∞(R4n,W4n), where W4n is the tensor product of three algebras, i.e., the hyperbolic quaternion B^, the bicomplex number B, and the Clifford algebra Rn. The operator D is a square root of the Laplacian in R4n, introduced by the formula D|=∑j=03Kj?Zj with Kjbeing the basis of B^, and ?Zj denoting the twisted Hermitian Dirac operators in the bicomplex Clifford algebra B?R0,4n whose definition involves a delicate construction of the bicomplexWitt basis. The introduction of the operator D can also overturn the prevailing opinion in the Hermitian Clifford analysis in the complex or quaternionic setting that the complex or quaternionic Hermitiean monogenic functions are described by a system of equations instead of by a single equation like classical monogenic functions which are null solutions of Dirac operator. In contrast to the Hermitian Clifford analysis in quaternionic setting, the Poisson brackets of the twisted real Clifford vectors do not vanish in general in the bicomplex setting. For the operator D, we establish the Cauchy integral formula, which generalizes the Martinelli-Bochner formula in the theory of several complex variables.     

A limiting relation between Chebyschev polynomials with a discrete argument and Legendre polynomials

Asymptotic estimates are obtained in a uniform metric and in the L _p metrics (p 2) for the difference between Chebyschev polynomials with a discrete argument and Legendre polynomials, under simultaneous passage to infinity of the degree of the polynomials and the number of lattice nodes at which the Chebyschev polynomials are defined.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 37–43, 1989.     

Difference equations for discrete classical multiple orthogonal polynomials

For discrete multiple orthogonal polynomials such as the multiple Charlier polynomials, the multiple Meixner polynomials, and the multiple Hahn polynomials, we first find a lowering operator and then give a (r+1)th order difference equation by combining the lowering operator with the raising operator. As a corollary, explicit third order difference equations for discrete multiple orthogonal polynomials are given, which was already proved by Van Assche for the multiple Charlier polynomials and the multiple Meixner polynomials.