Some polynomials defined by generating functions and differential equations

Classical orthogonal Polynomials — the Jacobi, Laguerre and Hermite polynomials — form the simplest class of special functions. At the same time, the theory of these polynomials admits wide generalizations. By using the Rodrigues formula for the Jacobi, Laguerre and Hermite polynomials we can come to integral representations for other special functions of mathematical physics, for example, hypergeometric functions and Bessel functions [N16, N18]. On the other hand, a construction scheme for the theory of these polynomials can naturally be generalized to classical orthogonal polynomials of a discrete variable. In view of this in Chap. 1 we shall give in a coherent way a brief description of some basic facts of the theory of classical orthogonal polynomials.

The subject of the study is the development of an analytical method for solving boundary value problems of a variant of the mathematical theory of transversally isotropic plates of arbitrary constant thickness, which boil down to the integration of systems of inhomogeneous high-order differential equations of equilibrium. According to the developed version of the theory, all components of the stress-strain state and boundary conditions are considered functions of three coordinates. The indicated functions of the three variables are expanded into infinite mathematical series by Legendre polynomials from the transverse coordinate. The boundary conditions on the front faces of the plates are fulfilled exactly. The boundary conditions on the lateral surfaces are fulfilled according to the appropriate approximation, which is determined by a certain number of terms in partial sums of mathematical series. This makes it possible to effectively determine all components of the stress-strain state with any high accuracy. It is also important to note that the developed version of the mathematical theory takes into account vortex and potential edge effects with high accuracy. The problem here lies in the high-order systems of differential equilibrium equations with partial derivatives. This complicates their solution. Moreover, the order of systems increases with an increase in the number of members in the partial mathematical sums of the development of components of the stress-strain state into infinite mathematical series. Therefore, the solution of this problem is connected with the application of a new methodology for finding partial and general solutions. The methodology consists in the fact that the initial systems of high-order differential equations of equilibrium are reduced by various mathematical transformations to convenient homogeneous and heterogeneous systems of high-order differential equations. These equations, in turn, are reduced to homogeneous and inhomogeneous differential equations of the second order by the developed operator method. The general and partial solutions of the initial equilibrium systems are expressed through the general and partial solutions of the second-order differential equations. The skew-symmetric transverse load of the plates relative to the median plane is considered. The goal is to obtain analytical solutions of boundary value problems for plates of arbitrary constant thickness. General solutions in special functions for components of the stress-strain state (SSS) from the annular transverse load of circular and annular plates under axisymmetric deformation are obtained. Analytical solutions of the boundary value problems for the specified plates under the action of axisymmetric loads for various static and kinematic boundary conditions on the lateral surfaces were obtained. An analysis of the obtained results was carried out.The proposed methodology, the method of solving systems of high-order differential equations of equilibrium, the method of obtaining general and partial solutions can also be applied in classical and refined theories, including theories of the Tymoshenko-Reissner type.

A very general set of orthogonal polynomials with five free parameters is given explicitly, the orthogonality relation is proved and the three term recurrence relation is found.

Lagrangian coordinates in free boundary problems for parabolic equations

Representations of orthogonal polynomials

When an attempt is made to construct an even particular solution of Mathieu's differential equation (written in a slightly modified notation), either with period 2π or with period 4π, by inserting into it an appropriate trigonometric series of the cosine type, it appears that the coefficients of this series satisfy a recurrence relation of a kind encountered in the theory of orthogonal polynomials. After splitting off a certain proportionality factor in each coefficient, there ultimately result two infinite sequences of parameter-dependent orthogonal polynomials constituting, respectively, a generalization of the Chebyshev polynomials of the first kind {Tn(x)} and a generalization of another special case of the Jacobi polynomials, namely {P(1/2,−1/2)n(x)}. The weight function corresponding to each of these orthogonal sequences consists of an infinite series of Dirac δ-peaks (mass-points) whose locations and (positive) coefficients are given by quantities appearing in the standard theory of Mathieu functions. With the aid of the author's theory of recursive generation of systems of orthogonal polynomials, each of the two orthogonal sequences may be complemented with infinitely many systems of associated orthogonal polynomials whose recurrence relations present a positive integer shift on the discrete independent variable compared to the recursion of the orthogonal sequence from which they originate. The treatment of Mathieu's differential equation with trigonometric series of the sine type, in view of constructing an odd particular solution with 2π or 4π as period, does not produce any sequence of orthogonal polynomials which is in essence not already comprised in those previously obtained.

These lectures contain a brief introduction to orthogonal polynomials with some applications to nonlinear recurrence relation. We discuss the spectral theory of orthogonal polynomials. We also mention other applications which may not be well known to the integrable systems community. For example we treat birth and death processes, the J-matrix method, and inversions of large Hankel matrices. The spectral theory of the Askey–Wilson operators on bounded and unbounded intervals is briefly mentioned.

It is well-known that a second-order differential equation with three regular singularities can be reduced to the Gaussian hypergeometric equation. A documented extension is the generalized hypergeometric equation, which also has three regular singularities, but is a differential equation of order higher than the second. In this paper a distinct generalization of the Gaussian hypergeometric equation is introduced (section 1), namely the extended hypergeometric equation, which is of second-order, and has two regular and one irregular singularity. It can be shown (Section 2) that it includes the Mathieu equation. The solution in power series, and with logrithmic singularities, are obtained in the neighborhood of the two regular singularities, as for the Gaussian type, with the diffrence (section 3) that the recurence formulas for the coefficients are not two-term but rather multiple-term. The solutions in the neighborhood of the irregular singularity at infinity are obtained (setion 4) by three methods, viz. normal integrals, Laurent series and transformation to Hill's equation. As a conclusion a physical problem is mentioned (section 5) in which the extended hypergeometric differential equation arises.

The solution of several instances of the Schrodinger equation (1926) is made possible by using the well-known orthogonal polynomials associated with the names of Hermite, Legendre and Laguerre. A relativistic alternative to this equation was proposed by Dirac (1928) involving differential operators with matrix coefficients. In 1949 Krein developed a theory of matrix-valued orthogonal polynomials without any reference to differential equations. In Duran A J (1997 Matrix inner product having a matrix symmetric second order differential operator Rocky Mt. J. Math. 27 585–600), one of us raised the question of determining instances of these matrix-valued polynomials going along with second order differential operators with matrix coefficients. In Duran A J and Grunbaum F A (2004 Orthogonal matrix polynomials satisfying second order differential equations Int. Math. Res. Not. 10 461–84), we developed a method to produce such examples and observed that in certain cases there is a connection with the instance of Dirac's equation with a central potential. We observe that the case of the central Coulomb potential discussed in the physics literature in Darwin C G (1928 Proc. R. Soc. A 118 654), Nikiforov A F and Uvarov V B (1988 Special Functions of Mathematical Physics (Basle: Birkhauser) and Rose M E 1961 Relativistic Electron Theory (New York: Wiley)), and its solution, gives rise to a matrix weight function whose orthogonal polynomials solve a second order differential equation. To the best of our knowledge this is the first instance of a connection between the solution of the first order matrix equation of Dirac and the theory of matrix-valued orthogonal polynomials initiated by M G Krein.

In a first course in differential equations the basic theme consists in obtaining explicitly the complete (i.e., general) solutions of well-known types of ordinary differential equations. Among these types one would certainly like to include the linear nth-order equation with constant coefficients not only because of its simplicity but also because of its common occurrence in engineering and other sciences. However, in asserting that the so called complementary function (CF) is indeed the general solution of the homogeneous equation, some textbooks either appear to pay insufficient attention to the significance of the statement or tend to confuse the issue and often simply point to the fact that the CF contains n arbitrary constants [see, e.g., 5, 9, 10]. Some authors who recognize this as inadequate even in a first course include in their books a statement of the basic uniqueness theorem for the associated initial value problem with or without a proof in a later chapter or an apology that the proof is beyond the scope of a first course [1, 2, 3, 4, 6, 7, 8]. Either way, the situation is not very satisfactory. One purpose of this note is to indicate a simple and elementary approach by which the problem can be overcome rather effortlessly. This is done in Part I, which is presented without skipping any of the details of calculation in order to show that it is not only complete and rigorous but is even within the reach of students who have no familiarity whatever with differential equations. Another matter this note deals with concerns the question of how to obtain a particular integral of L(y) =f where L is a linear differential operator. Here the standard methods, available after determining the CF, are (1) the annihilator method of undetermined coefficients, which succeeds if L has constant coefficients and f is such that there can be found a linear differential operator M with constant coefficients satisfying M(f) = 0, and (2) Lagrange's method of variation of parameters whose applicability is not limited as in (1). Part II describes a method somewhat different from these two that applies regardless of whether or not L has constant coefficients and irrespective of the nature of f. More importantly, this method does not depend on the predetermination of the CF. However, the applicability of this method is limited to those equations in which L can be expressed as a product of first order factors raised to appropriate powers. First I prove a simple basic lemma regarding certain indefinite integrals and then proceed directly to the theorem that gives the general solution of the homogeneous equation with constant coefficients. Only mathematical induction is used in the proofs in addition to the fundamental theorem of algebra that of course is, as always, assumed. Next I indicate an alternative and completely direct approach to the solution of the nonhomogeneous equation by repeated application of the exponential shift technique. When it applies, this method gives not merely a particular integral but in fact the general solution without any need to rely upon the fundamental uniqueness theorem for its justification.

On Some Simple Floquet Theory on Time Scales. On a Condition for Transitivity of Lorenz Maps. Iterated Multifunction Systems. Invariant Manifolds as Pullback Attractors of Nonautonomous Difference Equations. Determination of Initial Data Generating Solutions of Bernoulli's Type Difference Equations with Prescribed Asymptotic Behaviour. Solutions bounded on the Whole Line for Perturbed Difference Systems. On Rational Third Order Difference Equations. Decaying Solutions for Difference Equations with Deviating Argument. On a Differential and Difference Equation with a Constant Delay. Some Discrete Nonlinear Inequalities and Applications To Boundary Value Problems. The Asymptotic Stability of x(n + k) + ax(n) + bx(n _ l ) = 0. On Nonoscillatory Solutions of Third Order Difference Equations. Global Stability of Periodic Orbits of Nonautonomous Difference Equations in Population Biology and the Cushing-Henson Conjectures. Symplectic Factorizations and the Definition of a Focal Point. Second Smaller Zero of Kneading Determinant for Iterated Maps. Bifurcation of Almost Periodic Solutions in a Difference Equation. The Harmonic Oscillator { An Extension Via Measure Chains. Solution of Dirichlet Problems with Discrete Double-Layer Potentials. Moments of Solutions of Linear Difference Equations. Time Variant Consensus Formation in Higher Dimensions. On Asymptotic Properties of Solutions of the Difference Equation Dx(t) = -ax(t) + bx(g (t)). Oscillation Theorems for a Class of Fourth Order Nonlinear Difference Equations . 193. Iterates of the Tangent Map -- the Bifurcation Scheme. Difference Equations for Photon-Number Distribution in the Stationary Regime of a Random Laser. Delay Equations on Measure Chains: Basics and Linearized Stability. On the System of two Difference Equations xn+1 = p+yn-k / yn, yn+1 = q+xn-k / xn. Asymptotic Behavior of Solutions of Certain Second Order Nonlinear Difference Equations. Asymptotic Behavior of the Solutions of the Fuzzy Difference Equation . On the Gauss Hypergeometric Series with Roots Outside the Unit Disk. Symbolic Dynamics Generated by an Idealized Time-delayed Chua's Circuit.

Special issue: One hundred years of PVI, the Fuchs–Painlevé equation

The main aim of this paper is the study of the general solution of the exceptional Hermite differential equation with fixed partition $\lambda = (1)$ and the construction of minimal surfaces associated with this solution. We derive a linear second-order ordinary differential equation associated with a specific family of exceptional polynomials of codimension two. We show that these polynomials can be expressed in terms of classical Hermite polynomials. Based on this fact, we demonstrate that there exists a link between the norm of an exceptional Hermite polynomial and the gap sequence arising from the partition used to construct this polynomial. We find the general analytic solution of the exceptional Hermite differential equation which has no gap in its spectrum. We show that the spectrum is complemented by non-polynomial solutions. We present an implementation of the obtained results for the surfaces expressed in terms of the general solution making use of the classical Enneper-Weierstrass formula for the immersion in the Euclidean space $\mathbb{E}^3$, leading to minimal surfaces. Three-dimensional displays of these surfaces are presented.

Finding closed form solutions differential equations has a long history computer algebra. For example, the Risch algorithm (1969) decides if the equation y' = f can be solved terms elementary functions. These are functions that can be written terms exp and log, where in terms of allows for field operations, composition, and algebraic extensions. More generally, functions are closed form if they are written terms commonly used functions. This includes not only exp and log, but other common functions as well, such as Bessel functions or the Gauss hypergeometric function. Given a differential equation L, to find solutions written terms such functions, one seeks a sequence transformations that sends the Bessel equation, or the Gauss hypergeometric equation, to L. Although random equations are unlikely to have closed form solutions, they are remarkably common applications. For example, if y = ∑n=0∞ an xn has a positive radius convergence, integer coefficients an, and satisfies a second order homogeneous linear differential equation L with polynomial coefficients, then L is conjectured to be solvable closed form. Such equations are common, not only combinatorics, but physics as well. The talk will describe recent progress finding closed form solutions differential and difference equations, as well as open questions.

Introduction: Classical orthogonal polynomials such as the Legendre, Laguerre, Hermite, and Chebyshev polynomials have a wide range of applications across various domains of science and engineering. Aim: The aim of this research paper is to present selected recent developments in the theory of orthogonal polynomials, with particular emphasis on their analytical properties and their role in approximation theory. Methods: The study employs a combined quantitative-qualitative research methodology. It is based on the analysis of relevant scientific literature, from which data significant to the subject matter were collected, interpreted, and systematically examined. Results: The results indicate that contemporary research in probability theory, graph theory, coding theory, and related areas increasingly relies on the theory of orthogonal polynomials. This paper provides a theoretical analysis of the connection between orthogonal polynomials and their applications in specific computational problems. Fundamental properties of these polynomials are presented and illustrated through selected examples that contribute to a clearer understanding of their structure and practical relevance. Conclusion: The approximation of the transfer function of a low-pass filter can be achieved through a straightforward adaptation of orthogonal Jacobi polynomials. Furthermore, Gaussian quadrature represents a powerful numerical technique for the approximation of definite integrals, utilizing optimally chosen nodes and weight functions to achieve high accuracy with minimal computational effort. Orthogonal polynomials thus serve as a significant link between pure mathematics and engineering disciplines, highlighting the importance of further study on their properties and applications.