Difference Equations, Special Functions and Orthogonal Polynomials

PREFACE.

Contents.

Although the Pascal matrix is one of the oldest in the history of Mathematics, owing to both its utility in many applications and its countless properties, it continues to create interest. In this paper we review some recent works on the Pascal matrix by focusing our attention on its relations with linear algebra, difference equations and classical polynomials, such as the Legendre, Bernestein and Laguerre polynomials.

In the context of expanding piecewise monotone interval maps, we present an alternative proof of a main result of Baladi and Ruelle concerning weigthed zeta functions and weigthed kneading determinants.

The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a quasilinear Sobolev or pseudo-parabolic equation with initial jump. We have derived a method based on using finite elements with piecewise linear functions in space and with exponential functions in time and appropriate quadrature formulae with remainder term in integral form. In the initial layer, we introduce a special non-uniform mesh which is constructed by using estimates of derivatives of the exact solution and the analysis of the local truncation error. For the time integration we use the implicit rule. The fully discrete scheme is shown to be convergent of order 2 in space and of order one in time, uniformly in the singular perturbation parameter. Numerical results supporting the theory are presented.

We study the existence of 3-cycles (global periodicity) for the diiference equation x_n+2 = f(x_n+1, x_n), where f is a continuous map from (0,∞) × (0,∞) into (0,∞) and the initial conditions are positive real numbers. We use strongly monotonicity properties of fiber maps obtained from f.

We investigate a refined growth scale, logarithmic growth, for indeterminate moment problems of order zero. We show that the four entire functions appearing in the Nevanlinna parametrization have the same logarithmic order and type. In the appendix it is shown that the logarithmic indicator is constant.

In this paper we present a sequence of bi-orthogonal trigonometric polynomials, in the Szegő's sense, for which we study the usual topics in the orthogonal polynomial (OP) theory. In particular we obtain a new connection with OP on the unit circle and nice properties like: a five term recurrence relation with a Jacobi-type representation, a Favard's type theorem and a Christoffel-Darboux type formula.

In this paper, we express explicitly the linearization coefficients between three polynomial sets of Sheffer type using their corresponding lowering operators. We obtain some well known results as particular cases. A Crofton-type formula is derived. As application, we give linearization coefficients for Gould-Hopper polynomials as well as some reduction formulae for Kampé de Fériet functions.

In this paper, we characterize all d-orthogonal polynomial sets generated by functions of the type , where A and C are two formal power series satisfying A(0) = C′(0) = 1 and C(0) = 0. We, also, determine the corresponding d-dimensional functional vector of the obtained polynomial sets and we produce the d-orthogonal Chebyshev polynomials of the first kind.

In Hermite interpolation the problem of finding a polynomial P of degree at most (q + 1)n − 1 such that

has a unique solution (is regular) for arbitrary data and given—pairwise distinct—interpolation nodes z_i.

When the orders of derivatives j are replaced by k_j with 0 = k₀ < k₁ < ⋯ < k_q (so called Hermite–Birkhoff or lacunary interpolation), a general regularity property as above does not hold.

This paper will address two types of lacunary interpolation: there is one jump in the orders of derivatives only, either in the first (the case of (0, m, m + 1, …, m + r) interpolation) or in the last entry (the case of (0, 1, …, r, r + m) interpolation).

We describe the family of symmetric semiclassical linear functionals of class s = 2. In this way, some known linear functionals appear as well as a new one. We focus our attention in the integral representation of them using the corresponding Pearson equation.

This paper is a survey of the numerous Sierpinski curve Julia sets that arise in the family of rational maps given by

While these Julia sets are all the same from a topological point of view (they are all homeomorphic), the dynamics on these sets are almost always very different in the sense that no two maps are topologically conjugate.

This contribution is devoted to a discussion of the asymptotic behavior of solutions of systems of three difference equations. We try to connect two techniques – the so called retract type technique and Liapunov type approach. We show that under appropriate conditions containing conditions typical for the retract technique approach, and conditions typical for the Liapunov type approach, there exists at least one solution of the system considered the graph of which stays in a prescribed domain.

Using certain summation inequalities, the coexistence of various types of nonoscillatory solutions for an Emden-Fowler type difference equation is investigated. Discrepancies between discrete and continuous cases are pointed out as well.

We present the new concept applied for the comparison of conjoined bases of symplectic difference systems. This concept is closely related to the concept of multiplicities of focal points introduced by W. Kratz. Thus, the comparative index μ(i) = μ₁(i) + μ₂(i) gives us possibility to describe the case when the image condition and the Riccati inequality for conjoined bases Y_i, do not hold. In this work we prove the main properties of μ(i) and investigate connections between μ(i) and the number of focal points m(i).

In this survey I will discuss a q–umbral method reinvented by the author, which also involves the Nalli–Ward-Alsalam q–addition and the Jackson–Hahn–Cigler q–addition. This kind of q-calculus has a close connection to the Jackson q-gamma function and the Heine Ω function. In this way we can find q-analogues of the results of Nörlundfor generalized q-Bernoulli, q-Euler and q-Lucas numbers and polynomials, which were anticipated by Ward. This calculus has previously been used by P. Appell 1879, Ashton & Lindemann 1909, Smith & Pringsheim 1911 and Daum 1941. The Jacobi-Neville elliptic functions can easily be expressed in terms of the Heine Ω function and its tilde version. The advantages of this umbral method have been summarized in [10, p. 495].

This survey paper reports on the properties of the fourth-order Bessel-type linear ordinary differential equation, on the generated self-adjoint differential operators in two associated Hilbert function spaces, and on the generalisation of the classical Hankel integral transform. These results are based upon the properties of the classical Bessel and Laguerre second-order differential equations, and on the fourth-order Laguerre-type differential equation. From these differential equations and their solutions, limit processes yield the fourth-order Bessel-type functions and the associated differential equation.

We consider discrete laplacians for iterated maps on the interval and examine their eigenvalues. We have introduced a notion of conductance (Cheeger constant) for a discrete dynamical system, now we study their relations with the spectrum. We compute the systoles and the first eigenvalue of some families of discrete dynamical systems.

When one considers a linear differential equation

one is usually interested in solving this equation. But what does solve mean ? Is it to compute one solution or a basis of solutions ? And more precisely, is it to compute a numerical approximation of the solutions or to find a global form ? In this paper, we focus on closed form the solutions i.e. solutions that can be fully expressed using the usual functions. For example, one considers rational solutions, i.e., solutions in the field that defines the coefficients of the equation, since this problem is the starting point of many others algorithms like the factorization of linear differential operators or the computation of liouvillian solutions of linear differential equations (solutions which can be built up from the rational functions by algebraic operations, by taking exponentials and by integration). If many works were done on some special linear differential equations, the study of the general case is uneasy. It usually leads to heavy computations. So, the use of computer algebra is natural for solving such problems. But if the theoretical approach is well known, the algorithmic methods are recent, even for rational solutions of linear differential equations with coefficients in C(x) or in exponential extensions, and furthermore for more general solutions like solutions involving special functions.

In this paper, we summarize actual algorithms to compute closed form solutions of linear differential equations. Generalizations to linear differential systems and linear difference equations are briefly discussed.

In this paper the differential-difference systems with quadratic right-hand sides are considered. Conditions for stability, estimation of domain of stability and estimation of rate of convergence of solutions were given.

Characteristic Lie algebra of a discrete equation is studied. An algorithm of finding the algebra for a given equation is discussed.

Sufficient conditions for the global attractivity of the equilibrium point of the delay difference equations, which appears as models of the actions of neurons with dynamical threshold effects, are obtained by applying the method of Liapunov functionals and others.

In this paper we consider a bounded time scale 𝕋 = [a, b], a quadratic functional defined over such time scale, and its perturbation , where the endpoints of are zero, while the initial endpoint x(α) of can vary and x(b) is zero. It is known that there is no restriction on x(a) in when studying the positivity of these functionals. We prove that, when studying the nonnegativity, the initial state x(a) in must be restricted to a certain subspace, which is the kernel of a specific conjoined basis of the associated time scale symplectic system. This result generalizes a known discrete-time special case, but it is new for the corresponding continuous-time case. We provide several examples which illustrate the theory.

Monotonicity of dynamical systems is proven to be a very powerful means to discover long term behaviour of continuous-time and discrete-time dynamical systems. In this paper we consider monotone continuous-time dynamical systems and their discretizations with Runge-Kutta methods. We study the parameters of the discretization method that guarantee monotone discrete-time system as a result. In case of a general class of polyhedral order cones we conclude a simple and practically useful formula for the step sizes that ensure discrete monotonicity.

A linear difference-differential system with constant coefficients and containing a small parameter is investigated. We establish sufficient conditions for the existence of an asymptotic integral manifold of solutions described by a linear system of differential equations without delays.

We consider a discrete-time economic model which is a particular case of the Kaldor-type business cycle model and it is described by a two-dimensional dynamical system. Under certain conditions the map can be reduced to a skew map whose components, the base and the fiber map, both have entropy. Our proposal is to study and measure the complexity of the system using symbolic dynamics techniques and the topological entropy.

We expand a discrete–time lattice sine–Gordon equation on multiple lattices and obtain the partial difference equation which governs its far field behaviour. Such reduction allow us to obtain a new completely discrete nonlinear Schröedinger (NLS) type equation.

We compare the stability domains in the space of the parameters for the pare of differential and difference equations and x_n − x_n−1 = Ax_n−k in ℝ^m, as well as the pair of scalar equations with two delays and x_n − x_n−1 = ax_n−m + bx_n−k.

In this article we will show how computer algebra can be used in the study of orthogonal polynomials and special functions. The classical orthogonal polynomials named after Jacobi, Gegenbauer, Chebyshev, Legendre, Laguerre, Hermite and Bessel can be classified as the polynomial solutions of second order differential equations. Similarly the classical "discrete" orthogonal polynomials named after Hahn, Krawtchouk, Meixner and Charlier are classified as the polynomial solutions of second order difference equations. Using computer algebra one can compute the recurrence equations and hypergeometric representations of these systems, one can convert this process by computing differential and difference equations from the hypergeometric representations automatically, and one can decide whether a recurrence equation has classical orthogonal polynomial solutions. We will discuss these and related algorithms, and give some demonstrations with Maple.

We investigate convergence to a period-two solution for the second order difference equation

where all parameters α, β, γ, A, B, and C and initial conditions x₋₁, x₀ are non-negative, and such that A + Bx_n + Cx_n−1 > 0 for all n. Necessary and sufficient conditions for existence of prime period two solutions were given by Kulenović and Ladas. Here the authors study the basins of attraction of period-two solutions of (ε) for all cases where the right-hand-side has either two non-zero coefficients in both numerator and denominator, or one non-zero coefficients in numerator and three non-zero coefficients in the denominator. More precisely, it is shown that for these cases, a map associated with the equation has a unique saddle point in the positive quadrant, and it has two periodic orbits. The latter have large basins of attraction consisting of subsets of the positive quadrant of initial conditions in many cases separated by the stable manifold of the unique saddle point, which is the graph of a continuous monotonic function. Answers are given to three open problems posed by Kulenović and Ladas.

The first normal ordering problem involves bosonic harmonic oscillator creation and annihilation operators (Heisenberg algebra). It is related to the problem of finding the finite transformation generated by L_k−1 := −z^k ∂_z, k ∈ ℤ, z ∈ ℂ (conformal algebra generators). It can be formulated in terms of a subclass of Sheffer polynomials called Jabotinsky polynomials. The coefficients of these polynomials furnish generalized Stirling number triangles of the second kind, called S2(k;n,m) for k ∈ ℤ. Generalized Stirling-numbers of the first kind, S1(k;n, m) are also defined.

The second normal ordering problem appears in thermo-field dynamics for the harmonic Bose oscillator. Again Sheffer polynomials appear. They relate to Euler numbers and iterated sums of squares. In a different approach to this problem one solves the differential-difference equation f_n+1 = f'_n + n² f_n-1, n > = 1, with certain inputs f₀ and f₁ = f'₀.

In this case the integer coefficients of the special Sheffer polynomials which emerge have an interpretation as sum over multinomials for some subset of partitions.

Utilizing a result due to Krasnosel'skiĭ and Zabreiko, we establish the existence of non-trivial solutions for the family of even ordered dynamic equations

with the boundary conditions We will determine sufficient conditions on the function f to insure the existence of a solution.

We consider the Sobolev inner product

where (μ₀, μ₁) is a symmetrically coherent pair with one of the two measures the Hermite measure. We give a survey of the analytical properties of the corresponding Sobolev orthogonal polynomials and establish a new result about the asymptotic behaviour of these Hermite–Sobolev orthogonal polynomials inside the support of the measures μ₀ and μ₁.

The existence of nonoscillatory solutions with prescribed asymptotic behavior is studied for a nonlinear system of two coupled second order nonlinear equations. A general existence theorem for functional boundary value problems, stated by the authors in [1], is employed and the main advantages of this approach are illustrated.

In this paper we will classify the limits of the solutions of a linear delay difference system completely. In paticular, if the solution tends to an equiliblium point or a periodic orbit, we will give the explicit expressions in terms of the initial conditions and the delay parameters.

Recently it has been shown that multiple orthogonal polynomials are solutions of a Riemann-Hilbert problem. For the class of Generalized Nikishin systems we give the normalization at infinity for this Riemann-Hilbert problem.

We use a generalization of the Schwarz inequality to prove Turán-type inequalities for some special functions, such as the polygamma and the Riemann ζ functions. Finally we extend to x > 0 the range of validity of a lower bound for the function , proved by Elbert and Laforgia only for , p > 1. This function is related to gamma and incomplete gamma functions by .

A uniform asymptotics in the complex plane for the third Painlevé transcendent is constructed and proved. The leading term of asymptotics as |z| → ∞ is given by the Boutroux ansatz, i.e. by an elliptic function with its modulus depending on arg z. A functional equation for the modulus is universal for PIII equation and does not depend on initial conditions. It can be solved as an Abel problem of inversion of elliptic integrals. Another component of the Boutroux ansatz is the phase shift in the elliptic function. It depends on initial data, and we calculate it with the help of Isomonodromic Deformation Method (IDM). By solving a direct monodromy problem for a relevant Lax pair of operators, we fit given monodromy data with their approximations, coming from the leading term of asymptotics. This leads to explicit formula both for the modulus and the phase shift. Since a monodromy data for PIII transcendent can be expressed explicitly through the initial conditions at z = 0 (see ⁷), we come to the connection formulas linking the PIII transcendent asymptotics at infinity and at the origin. Finally, the IDM technique provides the proof of the above constructions, giving an analog of the Bolibrukh-Its-Kapaev theorem proved earlier for a similar asymptotic description of PII transcendent in the complex plane ⁹.

We consider families of dynamic equations x^Δ = f(t,x) with the initial condition x(t₀) = x₀ over different time scales, treating time scales as a parameter of such a family. Our goal is to understand the behavior of the solutions of such the same initial value problems over different time scales as bifurcations and limits over their underlying domains, i.e., the time scales. In a general setting, we show that the limit of solutions over convergent sequence of time scales converges to a solution over the limit time scale. A well known sequence of bifurcations occurs when f(t,x) = 4x(3/4 − x) between ℝ₊ with its asymptotic continuous solutions, and ℝ₊, with its chaotic solutions.

We show two results concerning approximation via "totally discrete" time scales. In the first, we show that any time scale can be approximated by a totally discrete time scale. In the second, we show the bifurcations in the parameter space for a simple two parameter family of dynamic equations—one parameter is the time scale itself, derived from Euler's method for solving differential equations. These results speak to the nature of numerical approximations.

We discuss about a generalization of the UC hierarchy which was recently introduced by Tsuda as an extension of the KP hierarchy. By generalizing the determinant expression of universal characters with the independent variable transformation given by Tsuda, the bilinear equations for the multicomponent determinants are constructed through the Laplace expansion technique. It is also shown that the periodic reductions derive discrete integrable 2+1 dimensional equations.

We extend the known results of the nonautonomous difference equation in the title to the situation where (i) the parameters β_n and γ_n are period-two sequences of nonnegative real numbers with γ_n not identically zero; (ii) the parameters A_n and B_n are period-two sequences of positive real numbers; and (iii) the initial conditions x₋₁ and x₀ are such that x₋₁,x₀ ∈ [0, ∞) and x₋₁ + x₀ ∈ (0, ∞).

Some integral comparison theorems for scalar second order linear differential equations are extended to dynamic equations on time scales. Several examples and applications to difference equations are given.

The aim of this note is to discuss the existence of nonoscillations for the difference system of mixed type

in terms of some matrix measures envolving the matrices P_i and Q_j (i = 1, …,ℓ, j = 1,…, m). Those measures are formulated in basis upon arbitrary logarithmic norms of matrices.

In this note we show that two-dimensional superintegrable model - nonisotropic oscillator with the ratio of frequency 2 : 1 generate both exactly and quasi-exactly solvable problems in one-dimensional quantum mechanics via separation of variables in Cartesian and parabolic coordinates. The work done in collaboration with E.Kalnins and W.Miller, Jr.

The invariant ergodic measures for generalized Boole type transformations are studied making use of the invariant quasi-measure approach, based on some special solutions to the Frobenius-Perron operator.

In this paper we study representations associated with the positive solution of a certain difference equation, with initial condition 0. These are representations that arise as Fock representations associated to a quadratic commutation relation. We define a space of parameters for these Fock representations and we determine the regions in these parameter space where the representations are bounded.

Better understanding of brain functions can have implications for the study of the dynamics of learning. We apply concepts and tools of nonlinear dynamics and chaos to the modelling and study of learning. We use the sinusoidal activation function to adjust the sinaptic weights of a neuron and we investigate the behaviour of the network when the learning rate changes. Our results show that these systems can exhibit a very rich dynamics, from regular to chaotic, and we calculate the topological entropy.

We investigate the asymptotic behavior of the α-th moment of the solution (X_n) of a stochastic difference equation with independent noises. Depending on α ∈ (0,1] and on the ratio u ↦ 2f(u)/g²(u) (where f is the intensity of the deterministic term and g is the intensity of the stochastic term), tends to 0 or to infinity. The analysis applies to a weak Euler-Maruyama approximation of a stochastic differential equation. To obtain our results we make use of an elementary lemma about the estimation of a positive continuous function from below by a positive continuous convex function.

We present a survey on the properties of the Bezout polynomials A(x) and B(x) solving the Bezout's problem A(x)P_n(x) + B(x)P'_n(x) = 1, when P_n(x) belongs to an orthogonal polynomial family. We extend results given by P. Humbert for Legendre polynomials on the several recurrences involving the four families P_n(x), P'_n(x), A(x) and B(x) and, from these recurrences, orthogonality of the Bezout pair (A(x),B(x)) is stated.

We present a novel analytical approach to the evaluation of the information entropy of Gegenbauer polynomials of parameter λ. This method allows us to express the entropy in terms of finite sums in the case λ ∈ ℕ. Furthermore, we obtain closed formulas when λ = 1, 2, 3.

Classical affine Lie algebras appear e.g. as symmetries of infinite dimensional integrable systems and are related to certain differential equations. They are central extensions of current algebras associated to finite-dimensional Lie algebras 𝔤. In geometric terms these current algebras might be described as Lie algebra valued meromorphic functions on the Riemann sphere with two possible poles. They carry a natural grading. In this talk the generalization to higher genus compact Riemann surfaces and more poles is reviewed. In case that the Lie algebra 𝔤 is reductive (e.g. 𝔤 is simple, semi-simple, abelian, …) a complete classification of (almost-) graded central extensions is given. In particular, for 𝔤 simple there exists a unique non-trivial (almost-)graded extension class. The considered algebras are related to difference equations, special functions and play a role in Conformal Field Theory.

We consider a class of higher order nonlinear neutral difference equations with quasidifferences where sequence (p_n) is a real sequence. The classification of nonoscillatory solutions of this equation are obtained. For a class of even order nonlinear neutral difference equations, dependent on (p_n), the conditions under which the eventually positive solutions of the equation can be classified into three nonempty distinct categories are given. We present sufficient condition under which the equation has a solution which converges to zero as well necessary and sufficient conditions under which the equation has a solution which tends to a nonzero constant and which diverges to infinity.

The critical group G of a connected graph is an abelian group, which is closely related to the discrete laplacian. The order of G is the number of spanning trees in the graph. Using the Markov partitions associated to the itinerary of the critical points of one-dimensional maps we define and study critical groups for m-modal discrete dynamical systems.

Ideal turbulence is a mathematical phenomenon which occurs in certain infinite-dimensional deterministic dynamical systems and implies that the attractor of a system lies off the phase space and among the attractor points there are fractal or even random functions. Ideal turbulence is observed in various idealized models of real distributed systems. Development of ideal turbulence is accompanied by cascade processes of birth of coherent structures of infinitely decreasing scales and this leads to certain problems in its visualization.

We review recent work on zeros of orthogonal polynomials.

We investigate the Lie point and generalized symmetries of certain nonlinear integrable equations on quad-graphs. Applications of the symmetry methods to such equations in obtaining group invariant solutions, related to discrete versions of the Painlevé differential equations, are also demonstrated.

We present some recent progresses on Heun functions, gathering results from classical analysis up to elliptic functions. We describe Picard's generalization of Floquet's theory for differential equations with doubly periodic coefficients and give the detailed forms of the level one Heun functions in terms of Jacobi theta functions. The finite-gap solutions give an interesting alternative integral representation which, at level one, is shown to be equivalent to their elliptic form.

We give four examples of families of orthogonal polynomials for which the coefficients in the recurrence relation satisfy a discrete Painlevé equation. The first example deals with Freud weights |x|^ρ exp(− |x|^m) on the real line, and we repeat Freud's derivation and analysis for the cases m = 2, 4, 6. The Freud equation for the recurrence coefficients when m = 4 corresponds to the discrete Painlevé I equation. The second example deals with orthogonal polynomials on the unit circle for the weight exp(λcosθ). These orthogonal polynomials are important in the theory of random unitary matrices. Periwal and Shevitz have shown that the recurrence coefficients satisfy the discrete Painlevé II equation. The third example deals with discrete orthogonal polynomials on the positive integers. We show that the recurrence coefficients of generalized Charlier polynomials can be obtained from a solution of the discrete Painlevé II equation. The fourth example deals with orthogonal polynomials on {±qⁿ : n ∈ ℕ}. We consider the discrete q-Hermite I polynomials and some discrete q-Freud polynomials for which the recurrence ceofficients satisfy a q-deformation of discrete Painlevé I.

The purpose of this paper is to extend some results of Karlin and McGregor's and Chihara's concerning the three-terms recurrence relation for polynomials orthogonal with respect to a measure on the nonnegative real axis. Our findings are relevant for the analysis of a type of Markov chains known as birth-death processes with killing.

Among boundary values problems (BVP) for partial differential equations there are certain classes of problems reducible to difference equations. Effective study of such problems has became possible in the last 20-30 years owing to appreciable advances done also in the theory of difference equations with discrete time, specifically given by one-dimensional maps. Here we apply how this reduction method may be used in simple nonlinear BVP, determined by a bimodal map. We consider two-dimensional linear hyperbolic system with constant coefficients, with nonlinear boundary conditions and usual initial conditions. The objective is to characterize the dependence of the motions of the vortice solutions with the topological invariants of the bimodal map.

By means of Abel's lemma on summation by parts, two functional equations with an extra integer parameter m will be established for Bailey's bilateral well-poised ₃ψ₃-series. Their limiting cases lead us directly to Bailey's non-terminating bilateral well-poised ₃ψ₃-series identities.

In this article, several 2+1 dimensional lattice hierarchies are constructed by using discrete operator zero curvature equation.