Clifford analysis versus its quaternionic counterparts

For a positive integer n let Cl_0,n be the universal Clifford algebra with the signature (0,n). The name Clifford analysis is usually referred to the function theories for functions in the kernels of the two operators: the (Cliffordian) Cauchy–Riemann operator and the Dirac operator. For n=2, Cl_0,2 becomes the skew‐field of Hamilton's quaternions for which the two operators are widely known: the Moisil–Théodoresco and the Fueter operators. We establish the precise relations between the Moisil–Théodoresco operator and the Dirac operator for Cl_0,3. It turns out that the case of the Cauchy–Riemann operator for Cl_0,3 and the Fueter operator is more sophisticated, and we describe the peculiarities emerging here. Copyright © 2009 John Wiley & Sons, Ltd.     

On the solutions of the Moisil–Théodoresco system

A structure theorem is proved for the solutions to the Moisil–Théodoresco system in open subsets of ?³. Furthermore, it is shown that the Cauchy transform maps L₂(?², ?_{0, 2}⁺) isomorphically onto H²(?₊³, ?_{0, 3}⁺), thus proving an elegant generalization to ?² of the classical notion of an analytic signal on the real line. Copyright © 2008 John Wiley & Sons, Ltd.     

The Gelfand–Tsetlin bases for Hodge–de Rham systems in Euclidean spaces

The main aim of this paper is to construct explicitly orthogonal bases for the spaces of k‐homogeneous polynomial solutions of the Hodge–de Rham system in the Euclidean space , which take values in the space of s‐vectors. Actually, we describe even the so‐called Gelfand–Tsetlin bases for such spaces in terms of Gegenbauer polynomials. As an application, we obtain an algorithm on how to compute an orthogonal basis of the space of homogeneous solutions for an arbitrary generalized Moisil–Théodoresco system in . Copyright © 2012 John Wiley & Sons, Ltd.     

On some constructive aspects of monogenic function theory in ℝ4

As is well known, a possible generalization to ?⁴ of the classical Cauchy–Riemann system leads to the so‐called Riesz system. The main goal of this paper is to construct explicitly a complete orthonormal system of polynomial solutions of this system with respect to a certain inner product. This will be done in the spaces of square integrable functions on the unit ball over ?. Copyright © 2011 John Wiley & Sons, Ltd.     

3D deformations by means of monogenic functions

In this paper, the authors compute the coefficient of quasiconformality for monogenic functions in an arbitrary ball of the Euclidean space . This quantification may be needed in applications but also appears to be of intrinsic interest. The main tool used is a 3D Fourier series development of monogenic functions in terms of a special set of solid spherical monogenics. Ultimately, we present some examples showing the applicability of our approach. Copyright © 2012 John Wiley & Sons, Ltd.     

Special functions and systems in Hermitian Clifford analysis: Higher codimension

In this paper, we investigate a Cauchy–Kowalevski (CK) extension problem that arises naturally within the framework of Hermitian Clifford analysis, which concerns the study of Dirac‐like operators in several complex variables. The work presented here includes CK extensions of higher codimension and in particular the CK extension of the Gauss distribution in several complex variables. Copyright © 2013 John Wiley & Sons, Ltd.     

Some families of generating functions for a class of bivariate polynomials

The main purpose of this paper is to present various families of generating functions for a class of polynomials in two variables. Furthermore, several general classes of bilinear, bilateral or mixed multilateral generating functions are obtained for these polynomials.     

Special functions and systems in Hermitian Clifford analysis

In this paper, we study some new special functions that arise naturally within the framework of Hermitian Clifford analysis, which concerns the study of Dirac‐like systems in several complex variables. In particular, we focus on Hermite polynomials, Bessel functions, and generalized powers. We also derive a Vekua system for solutions of Hermitian systems in axially symmetric domains. Copyright © 2012 John Wiley & Sons, Ltd.     

Reduction and transformation formulas for the Appell and related functions in two variables

In many seemingly diverse areas of applications, reduction, summation, and transformation formulas for various families of hypergeometric functions in one, two, and more variables are potentially useful, especially in situations when these hypergeometric functions are involved in solutions of mathematical, physical, and engineering problems that can be modeled by (for example) ordinary and partial differential equations. The main object of this article is to investigate a number of reductions and transformations for the Appell functions F₁,F₂,F₃, and F₄ in two variables and the corresponding (substantially more general) double‐series identities. In particular, we observe that a certain reduction formula for the Appell function F₃ derived recently by Prajapati et al., together with other related results, were obtained more than four decades earlier by Srivastava. We give a new simple derivation of the previously mentioned Srivastava's formula 12 . We also present a brief account of several other related results that are closely associated with the Appell and other higher‐order hypergeometric functions in two variables. Copyright © 2017 John Wiley & Sons, Ltd.     

Quadratic Hermite–Padé polynomials associated with the exponential function

The asymptotic behavior of quadratic Hermite–Padé polynomials associated with the exponential function is studied for n→∞. These polynomials are defined by the relation

(*)

p_n(z)+q_n(z)e^z+r_n(z)e^2z=O(z³ⁿ⁺²) as z→0,

where O(·) denotes Landau's symbol. In the investigation analytic expressions are proved for the asymptotics of the polynomials, for the asymptotics of the remainder term in (*), and also for the arcs on which the zeros of the polynomials and of the remainder term cluster if the independent variable z is rescaled in an appropriate way. The asymptotic expressions are defined with the help of an algebraic function of third degree and its associated Riemann surface. Among other possible applications, the results form the basis for the investigation of the convergence of quadratic Hermite–Padé approximants, which will be done in a follow-up paper.     

On the Faber coefficients of functions univalent in an ellipse

Let be the elliptical domain

Let denote the class of functions analytic and univalent in and satisfying the conditions and . In this paper, we obtain global sharp bounds for the Faber coefficients of the functions in certain related classes and subclasses of

     

On Multivariate Quasipolynomials of the Minimal Deviation from Zero

We generalize to several variables both the upper and the lower Gelfond bounds for the least uniform deviation from zero of the quasipolynomials (or Müntz–Legendre polynomials) on the segment [0, 1]. Orthonormal quasipolynomials are also considered.     

Two-point Taylor expansions in the asymptotic approximation of double integrals. Application to the second and fourth Appell functions

The main difficulty in Laplace's method of asymptotic expansions of double integrals is originated by a change of variables. We consider a double integral representation of the second Appell function F₂(a,b,b^′,c,c^′;x,y) and illustrate, over this example, a variant of Laplace's method which avoids that change of variables and simplifies the computations. Essentially, the method only requires a Taylor expansion of the integrand at the critical point of the phase function. We obtain in this way an asymptotic expansion of F₂(a,b,b^′,c,c^′;x,y) for large b, b^′, c and c^′. We also consider a double integral representation of the fourth Appell function F₄(a,b,c,d;x,y). We show, in this example, that this variant of Laplace's method is uniform when two or more critical points coalesce or a critical point approaches the boundary of the integration domain. We obtain in this way an asymptotic approximation of F₄(a,b,c,d;x,y) for large values of a,b,c and d. In this second example, the method requires a Taylor expansion of the integrand at two points simultaneously. For this purpose, we also investigate in this paper Taylor expansions of two-variable analytic functions with respect to two points, giving Cauchy-type formulas for the coefficients of the expansion and details about the regions of convergence.     

On the limit behavior of recurrence coefficients for multiple orthogonal polynomials

In this paper we investigate general properties of the coefficients in the recurrence relation satisfied by multiple orthogonal polynomials. The results include as particular cases Angelesco and Nikishin systems.     

Adaptive Fourier decomposition of functions in quaternionic Hardy spaces

We study decompositions of functions in the Hardy spaces into linear combinations of the basic functions in the orthogonal rational systems B_n, which are obtained in the respective contexts through Gram–Schmidt orthogonalization process on shifted Cauchy kernels. Those lead to adaptive decompositions of quaternionic‐valued signals of finite energy. This study is a generalization of the main results of the first author's recent research in relation to adaptive Takenaka–Malmquist systems in one complex variable. Copyright © 2011 John Wiley & Sons, Ltd.     

Ratio and relative asymptotics of polynomials orthogonal with respect to varying Denisov-type measures

Let μ be a finite positive Borel measure with compact support consisting of an interval plus a set of isolated points in , such that μ^′>0 almost everywhere on [c,d]. Let , be a sequence of polynomials, , with real coefficients whose zeros lie outside the smallest interval containing the support of μ. We prove ratio and relative asymptotics of sequences of orthogonal polynomials with respect to varying measures of the form dμ/w_2n. In particular, we obtain an analogue for varying measures of Denisov's extension of Rakhmanov's theorem on ratio asymptotics. These results on varying measures are applied to obtain ratio asymptotics for orthogonal polynomials with respect to fixed measures on the unit circle and for multi-orthogonal polynomials in which the measures involved are of the type described above.     

Dynamics of particle trajectories in a Rayleigh–Bénard problem

Fluid particle trajectories for the Rayleigh–Bénard problem in a cube with perfectly conducting lateral walls have been investigated. The velocity and temperature fields of the stationary flow solutions have been obtained by means of a parameter continuation procedure based on a Galerkin spectral method. The rich dynamics of the resulting fluid particle paths has been studied for three branches of stationary solutions and different values of the Rayleigh number within the range10⁴Ra1.5×10⁵ at a Prandtl number equal to 130. The stability properties and bifurcations of fixed points, which play a key role in the global dynamics, have been analyzed. Main periodic orbits and their stability character have also been determined. Poincaré maps reveal that regions of chaotic motion and regions of regular motion coexist inside the cavity. The boundaries of these three-dimensional regions have been determined. The metric entropy gives an indication of the mixing properties of the large chaotic zone.     

The use of the Sherman–Morrison–Woodbury formula to solve cyclic block tri-diagonal and cyclic block penta-diagonal linear systems of equations

The article presents a new theoretical viewpoint of Batista’s algorithms for solving cyclic tri-diagonal (and penta-diagonal) linear systems. The theory is based on the Sherman–Morrison–Woodbury formula.     

Inframonogenic functions and their applications in 3‐dimensional elasticity theory

Solutions of the sandwich equation , where stands for the first‐order differential operator (called Dirac operator) in the Euclidean space , are known as inframonogenic functions. These functions generalize in a natural way the theory of kernels associated with , the nowadays well‐known monogenic functions, and can be viewed also as a refinement of the biharmonic ones. In this paper we deepen study the connections between inframonogenic functions and the solutions of the homogeneous Lamé‐Navier system in . Our findings allow to shed some new light on the structure of the solutions of this fundamental system in 3‐dimensional elasticity theory.