Matrix-Valued hypergeometric Appell-Type polynomials

The family of q-Konhauser matrix polynomials have been extended to Konhauser matrix polynomials. The purpose of the present work is to show that an extension of the explicit forms, generating matrix functions, matrix recurrence relations and Rodrigues-type formula for these matrix polynomials are given, our desired results have been established and their applications are presented.

The main aim of this paper is to define and study of a new Horn’s matrix function, say, the p and q-Horn’s matrix function of two complex variables. The radius of regularity on this function is given when the positive integers p and q are greater than one, an integral representation of pHq 2 is obtained, recurrence relations are established. Finally, we obtain a higher order partial differential equation satisfied by the p and q-Horn’s matrix function.

Various families of generating matrix functions have been established in diverse ways. The objective of the present paper is to investigate these generalized Hermite matrix polynomials, and derive some important results for them, such as, the generating matrix functions, matrix recurrence relations, an expansion of xnI, finite summation formulas, addition theorems, integral representations, fractional calculus operators, and certain other implicit summation formulae.

Inversion components of block Hankel-like matrices

There are two definitions for the discrete q-Hermite polynomials, one of them is defined for \(0 1\). This paper is devoted to extend these definitions to the discrete q-Hermite matrix polynomials by means of the generating matrix functions. Explicit expressions and Rodrigues-type formulas for the discrete q-Hermite matrix polynomials are obtained. Some recurrence relations for these matrix polynomials, in particular the three terms recurrence relations are given. Furthermore, some identities are proved.

We study the solution of a system of higher-dimensional ordinary differential equations of block structure. Some separate subsystems are connected with each other by the nonseparated boundary conditions caused by an arbitrary relation between the boundary values of the solutions to the subsystems. For numerical solution, we propose some scheme of the method of transferring the boundary conditions that takes into account the specific characteristics of the systems under consideration. The results of numerical experiments are given. DOI: 10.1134/S1990478915010019 In this article, taking into account the block structure of the system of differential equations and the weak but arbitrary filling of the boundary condition matrix, we propose a version of the method of transfer of the boundary conditions. The advantage of this approach in comparison with the direct use of transfer methods in general form is obvious since here the transfer is carried out only with respect to those variables whose coefficients in the boundary conditions are nonzero; moreover, the transfer is carried out with the use of the only subsystem of differential equations involving the variable that is being transferred. We show the results of the numerical experiments obtained by solving a test problem based on the problem of calculating an unsteady fluid motion in a fragment of a pipeline network of complex structure.

Network is considered naturally as a wide range of different contexts, such as biological systems, social relationships as well as various technological scenarios. Investigation of the dynamic phenomena taking place in the network, determination of the structure of the network and community and description of the interactions between various elements of the network are the key issues in network analysis. One of the huge network structure challenges is the identification of the node(s) with an outstanding structural position within the network. The popular method for doing this is to calculate a measure of centrality. We examine node centrality measures such as degree, closeness, eigenvector, Katz and subgraph centrality for undirected networks. We show how the Katz centrality can be turned into degree and eigenvector centrality by considering limiting cases. Some existing centrality measures are linked to matrix functions. We extend this idea and examine the centrality measures based on general matrix functions and in particular, the logarithmic, cosine, sine, and hyperbolic functions. We also explore the concept of generalised Katz centrality. Various experiments are conducted for different networks generated by using random graph models. The results show that the logarithmic function in particular has potential as a centrality measure. Similar results were obtained for real-world networks.

In this chapter we consider matrix polynomials, i.e. matrix functions L(z) of the form \(L(z) = \sum\nolimits_{i = 0}^e {{A_i}{z^i},} \) where A 0 A l,..., A l are n × n matrices. We will assume throughout also that L(z) is regular, i.e. det L(z) does not vanish identically. One of our aims is to specify for matrix polynomials in a detailed way the theory developed in the first chapter for analytic matrix functions. We also solve here the first interpolation problem in this book, namely, how to construct a matrix polynomial with given null structure. Explicit formulas for the solution are obtained.

In this study, we examine integrals involving the incomplete H-function of several complex variables in combination with general class polynomials represented in product form. By employing definite integrals from well-known mathematical references, we derive closed-form results that highlight the interplay between special functions and polynomial structures. The incomplete H-function, known for its remarkable generality and applications in multivariate function theory, is here integrated with polynomial structures that unify many classical families. The integrals derived are expressed in a simplified and compact form, making them particularly suitable for applications in mathematical physics, probability theory, and statistical modeling. These results not only broaden the scope of incomplete special functions but also offer efficient tools for handling problems involving products of functions in multidimensional settings. Furthermore, the work provides a foundation for future investigations into more general classes of incomplete multivariable functions and their applications.

In 1967 Roger Penrose introduced what is now called twistor geometry (Penrose 1967). This geometry is an amplification of the geometry of lines in projective space studied extensively by Felix Klein and others in the late 19th century. Penrose made the fundamental observation that this classical geometry could be used in conjunction with a Radon transform style integral geometry to provide rather complete sets of solutions of numerous partial differential equations, both linear and nonlinear, which arise naturally in theoretical physics. The Radon transform, which involves integration over straight lines in R 3, was used by F. John to generate solutions of hyperbolic partial differential equations in 1942. Penrose integrates over complex projective lines, real 2-spheres, and the data of the integration is complex-analytic functions of 3 complex variables. This is analogous to (and indeed related to) Weierstrass' representation of solutions of the minimal surface equation in terms of triples of holomorphic functions of a single complex variable which satisfy an algebraic relation. In this paper we discuss the origins of the differential equations which are amiable to twistor-geometric study from the realms of theoretical physics. Twistor geometry is most suitable for physical theories which are relativistically invariant, including Maxwell's description of electromagnetic phenomena from the 19th century (Maxwell's equations), Einstein's theory of gravitation (the equations of general relativity), and the contemporary descriptions of elementary particles (quantum field theory). Twistor geometry interacts in a nontrivial manner with the mathematical models involved in all of the above physical theories. In particular, it has led to new solutions of some of the nonlinear partial differential equations involved (Einstein's equations, Yang-Mills equations, equations of a magnetic monopole). We will describe some recent progress on twistorgeometric representations of solutions of classical field equations. There are two aspects to this. First, there is the problem of translating a given system of differential equations into twistor-geometric language. This has been done successfully in many different cases, and will be described in Section 3. Once a given problem has been transferred to a twistor-geometric context, it becomes, in general, a specific problem involving algebraic geometry, several complex variables, and algebraic topology involving holomorphic data on complex manifolds. The only partial differential equations remaining in the twistor context are the CauchyRiemann equations characterizing holomorphic functions. In effect, the partial differential equation has been transformed into a geometric problem. This is analogous to the Fourier transform transforming a partial differential equation with constant coefficients into an algebraic problem, where differentiation is replaced by the (algebraic) symbol of the differential operator in question. The second aspect of the recent progress on the twistor-geometric representation of the solutions of the classical field equations concerns itself with solving the geometric problem in question and transforming back to space-time (or Euclidean space), generating (sometimes new) solutions of the classical field equations.

Orthogonal matrix polynomials and higher-order recurrence relations

This paper presents results on some boundary value problems for holomorphic functions of several complex variables in polydomains. The Cauchy kernel is one of the significant tools for solving the Riemann and the Riemann-Hilbert boundary value problems for holomorphic functions as well as for establishment of the connection between them. For polydomains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in polydomains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. The general integral representation formulas for the functions, holomorphic in polydomains, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holomorphic function for arbitrary polydomains is given and an exact, yet compact way of notation for holomorphic functions in arbitrary polydomains is introduced and applied. The Riemann jump problem and the Riemann-Hilbert problem are solved for holomorphic functions of several complex variables with the unit torus as the jump manifold. The higher-dimensional Plemelj-Sokhotzki formula for holomorphic functions in polydomains is established. The canonical functions of the Riemann problem for polydomains are represented and applied in order to construct solutions for both of the homogeneous and inhomogeneous problems. For all three boundary value problems, well-posed formulations are given which does not demand more solvability conditions than in the one variable case. The connection between the Riemann and the Riemann-Hilbert problem for polydomains is proven. Thus contrary to earlier research the results are similar to the respective ones for just one variable.

This paper contains further steps towards a Szegö theory for orthogonal rational matrix‐valued functions on the unit circle 𝕋. It continues the investigations started in [18]–[20]. Hereby we are guided by former work of Bultheel, González–Vera, Hendriksen, and Njåstad on scalar orthogonal rational functions on the one side and by research of Delsarte, Genin, and Kamp on matrix polynomials on the other side. In this paper we derive recursion formulas for Christoffel–Darboux pairs of rational matrix functions which lead us to jqq ‐recursively connected pairs of rational matrix functions. Moreover, we prove a Favard‐type theorem for rational matrix functions. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

In the article we obtained sufficient conditions of the existence of the nonlinear Noetherian boundary value problem solution for the system of differential-algebraic equations which are widely used in mechanics, economics, electrical engineering, and control theory. We studied the case of the nondegenerate system of differential algebraic equations, namely: the differential algebraic system that is solvable relatively to the derivative. In this case, the nonlinear system of differential algebraic equations is reduced to the system of ordinary differential equations with an arbitrary continuous function. The studied nonlinear differential-algebraic boundary-value problem in the article generalizes the numerous statements of the non-linear non-Gath boundary value problems considered in the monographs of А.М. Samoilenko, E.A. Grebenikov, Yu.A. Ryabov, A.A. Boichuk and S.M. Chuiko, and the obtained results can be carried over matrix boundary value problems for differential-algebraic systems. The obtained results in the article of the study of differential-algebraic boundary value problems, in contrast to the works of S. Kempbell, V.F. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and A.A. Boychuk, do not involve the use of the central canonical form, as well as perfect pairs and triples of matrices. To construct solutions of the considered boundary value problem, we proposed the iterative scheme using the method of simple iterations. The proposed solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem, were illustrated with an example. To assess the accuracy of the found approximations to the solution of the nonlinear differential-algebraic boundary value problem, we found the residuals of the obtained approximations in the original equation. We also note that obtained approximations to the solution of the nonlinear differential-algebraic boundary value problem exactly satisfy the boundary condition.

The main new results of this paper concern the formulation of algebraic conditions for the Fredholm property of elliptic systems of P.D.E.’s with boundary values, which are equivalent to the Lopatinskii condition. The Lopatinskii condition is reformulated in a new algebraic form (based on matrix polynomials) which is then used to study the existence of homotopies of elliptic boundary value problems. The paper also contains an exposition of the relevant parts of the theory of matrix polynomials and the theory of elliptic systems of P.D.E.’s.