Difference Equations, Special Functions and Orthogonal Polynomials - Proceedings of the International Conference

✍ Scribed by Jim M Cushing (editor), Saber N Elaydi (editor), Rupert Lasser (editor), Vassilis Papageorgiou (editor), Andreas Ruffing (editor), Walter Van Assche (editor)

• Publisher: World Scientific Publishing Company
• Year: 2007
• Tongue: English
• Leaves: 789
• Edition: Illustrated
• Category: Library

✦ Synopsis

This volume contains talks given at a joint meeting of three communities working in the fields of difference equations, special functions and applications (ISDE, OPSFA, and SIDE). The articles reflect the diversity of the topics in the meeting but have difference equations as common thread. Articles cover topics in difference equations, discrete dynamical systems, special functions, orthogonal polynomials, symmetries, and integrable difference equations.

✦ Table of Contents

Contents
Preface
Pascal Matrix, Classical Polynomials and Difference Equations L. Aceto and D. Tngiante
1. Introduction
2. Relevant properties
2.1. Connections with some known matrices
3. Classical polynomials
3.1. Bernstein and Legendre polynomials
3.2. Bernoulli and Euler polynomials
3.3. Hermite polynomials
3.4. Laguerre polynomials
4. Matrices, difference equations and polynomials
5. Conclusions
References
On the Ruelle Zeta Function of an Expanding Interval Map J. F. Alves and J. L. Fachada
1. Introduction
2. Proof of Theorem 1.3
Acknowledgments
References
Difference Schemes for the Singularly Perturbed Sobolev Equations G. M. Amiraliyev and I. G. Amiraliyeva
1. Introduction
2. Properties of the Solution of (1)-(3)
3. Discretization and Layer-adapted Mesh
3.1. Construction of the difference scheme
3.2. The Mesh
4. Uniform Error Estimates
4.1. Stability Bounds
5. Algorithm and Numerical Results
6. Conclusion
References
On Global Periodicity of x,+2 = f(x,+1, 2,) F. Balibrea and A . Linero
1. Introduction
2. Preliminary results for p = 3
3. The particular case f ( z , y ) = u ( z ) p ( y )
Bibliography
Logarithmic Order and Type of Indeterminate Moment Problems C. Berg and H. L. Pedersen (with an Appendix by W. Hayman)
1. Introduction and results
2. Proof of the main results
3. Stieltjes moment problems
4. Examples
5. The logarithmic growth scale
6. Appendix: The Phragmh-Lindeliif indicator of some functions of order zero
References
A System of Biorthogonal Trigonometric Polynomials E. Berriochoa, A . Cachafeiro and J. Garcia-Amor
1. Introduction
2. A sequence of trigonometric polynomials
3. Favard’s type theorem
References
Quasi Monomiality and Linearization Coefficients for Sheffer Polynomial Sets H. Chaggara
1. Introduction
2. Linearization coefficients
2.1. A general result
2.2. Shener polynomials
2.3. Generalized Hennite polynomials
2.4. Modified Laguerre polynomials
2.5. Orthogonal Shefler polynomials
3. Linearization coefficients for Appell PSs
3.1. General expression
3.2. Application to Gould-Hopper polynomials
References
d-orthogonal Polynomial Sets of Chebyshev Type Y. Ben Cheikh and N. Ben Romdhane
1. Introduction and preliminary results
2. Proof of the main result
3. Special cases
3.1. Case d=1.
3.2. Case d=2.
3.3. Case d 21.
4. General properties
4.1. A ( d + 1)-order recurrence relation
4.2. d -dimensional finctional
5. Chebyshev d-OPS of the first kind
References
On Two Problems in Lacunary Polynomial Interpolation M. G. de Bruin
1. Introduction
2. The case of (0, 1, . . . , T , T + m) interpolation
3. The case of (O,m, m + 1 , . . . , m + T ) interpolation
4. Proofs for Sec. 2
5. Proofs for Sec. 3
References
Semiclassical Linear F'unctionals of Class 2: The Symmetric Case A . M. Delgado and F. Marcelldn
1. Introduction
2. Some definitions and preliminary results
3. Construction of symmetric semiclassical linear functionals of class 2
References
A Myriad of Sierpinski Curve Julia Sets R. L. Devaney
1. Introduction
2. Preliminaries
3. Escape Sierpinski Curve Julia Sets
4. Buried Sierpinski Curves
5. Structurally Unstable Sierpinski Curves
6. Final Comments and Conjectures
References
Compulsory Asymptotic Behavior of Solutions of System of Three Discrete Equations J . Dabla% and I. R4ZiCkovd
1. Introduction
1.1. Description of the Problem Considered and Auxiliary Notions
1.2. Preliminary
1.2.1. Consequent point
1.2.2. Liapunov type polyfacial set
1.2.3. Points of strict egress and their geometrical sense
1.2.4. Retract and retraction
2. Main Result
2.1. Proof of the Main Result
2.1.1. Specification of the general scheme of the proof
2.1.2. Auxiliary mapping R1
2.1.3. Auxiliary mapping R:2
2.1.4. Resulting mapping R and its properties
Acknowledgments
References
Nonoscillatory Solutions for Emden-Fowler Type Difference Equations M. Cecchi, 2. Do&& M. Marini and I. VrkoC
1. Introduction
2. Main results
3. Discrete versus continuous case
References
The Comparative Index for Conjoined Bases of Symplectic Difference Systems J. V. Elyseeva
1. Introduction
2. Main results
3. Symplectic factorizations and index results
References
A Renaissance for a q-umbra1 Calculus T. Ernst
References
Fourth-order Bessel-type Special Functions: A Survey W. N. Everitt
1. Introduction
2. History
3. The fourth-order differential equation
4. Higher-order differential equations
5. The fourth-order differential expression LM
6. Hilbert function spaces
7. Differential operators generated by LM
8. Differential operators in L2((0, 00); z)
9. Self-adjoint operators in L2((0, 00); z)
10. Boundary properties at O+
11. Explicit boundary condition functions at O+
12. Spectral properties of the fourth-order Bessel-type operators
13. The F’riedrichs extension F
14. Self-adjoint operator s k in L2([0, 00); m k )
15. Spectral properties of the self-adjoint operator S k
16. Distributional orthogonality relationships
17. The generalised Hankel transform
18. The Plum partial differential equation
References
Laplacians and the Cheeger Constants for Discrete Dynamical Systems S. Fernandes, C. Gra'cio and J. Sousa Ramos
1. Introduction
2. Discrete Laplacian and conductance
3. Systoles in discrete dynamical systems
Acknowledgments
References
Algorithms Around Linear Differential Equations A . Fredet
1. Introduction
2. Rational and exponential solutions of linear differential equations
2.1. LDE with coemcients in C ( x )
2.2. LDE with coeficients in exponential extensions
3. Liouvillian solutions of linear differential equations
4. Solutions of linear differential equations in term of special functions
5. Linear differential systems
6. Linear difference equations
7. Conclusion
References
Stability of Differential-Difference Systems with Quadratic Right-Hand Sides D. Khysainov and I. Grytsay
1. Introduction
2. Preliminary
3. Main results
3.1. Linear case
3.2. Nonlinear case
References
Characteristic Algebras of Discrete Equations I. Habibullin
1. Introduction
2. Invariants and vector fields
3. Computation of the characteristic algebra
4. Characteristic algebra for the discrete Liouville equation
5. How to find the invariants?
6. Conclusion
Acknowledgments.
References
On the Asymptotic Behaviour of Solutions of Neuronic Difference Equations Y. Hamaya
1 Introduction
2 Global Attractivity of Eq.(l) and Others
Acknowledgements
References
Perturbation of Nonnegative Time Scale Quadratic Functionals R. Hilscher and V. R4.ZiEkova'
1. Introduction and Motivation
2. Main Results - Nonnegativity
3. Positive Definiteness
4. Continuous-Time Case
References
Monotonicity of Dynamical Systems and Their Discretizations 2. Horva'th
1. Motivation
2. Preliminaries; continuous-time problems
3. Discrete-time problems
3.1. Time discretization of with Runge-Kutta methods
4. Discrete monotonicity of RK methods
4.1. Discrete monotonicity of the explicit Euler method
References
Construction of an Integral Manifold for a Linear Delay System K. R. Janglajew
1. Introduction and preliminaries
2. Successive approximations
3. Convergence
References
Chaotic Behavior in a Two-dimensional Business Cycle Model C. Janua'rio, C. Gra'cio and J. Sousa Ramos
1. Introduction
2. The model and some considerations about triangular maps
3. Chaotic behavior in the map F
4. Final conclusions
References
Multiscale Expansion of the Lattice sine-Gordon Equation X.-D. Ji, D. Levi and M. Petrera
1. Introduction
2. The discrete perturbation technique
3. Reduction of the lattice sine-Gordon equation
Acknowledgments
References
Stability of Delay Difference and Differential Equations: Similarities and Distinctions M. M. Kipnis and I. S. Levitskaya
1. Equation k ( t ) = a z ( t - 7 ) V8 Equation Zn - Zn-1 = UZn-k Euler - Levin and May (1976)
2. System &(t) = A s ( t - T ) US System xn - xn-1 = A5n-k Rekhlitskii (1956) - Levitskaya (2005)
3. Equation $(t) = a x ( t - 7 1 ) 4- b x ( t - 7 2 ) US Levitskaya (2005) - Kipnis and Levitskaya (2005) Equation X n - X n - i = a Xn-m + b X n - k
3.1. Preliminaries
3.2. Results
3.3. Explicit Asymptotic Stability Conditions Gopalsamy (1 990), Vaguina and Kipnia (2003), Cook and Gyori (1990), Kipnis and Levitskaya (2005)
3.4. Summary
References
Computer Algebra Methods for Orthogonal Polynomials W. Koepf
1. Demonstrations with Computer Algebra
2. Classical Orthogonal Polynomials
3. Hypergeometric Functions
4. Computation of the Recurrence Coefficients
5. Zeilberger’s Algorithm
6. Petkovsek-van Hoeij Algorithm
7. Recurrence Operators
8. Classical Orthogonal Polynomial Solutions of Recurrence Equations
9. Final Remarks
References
Convergence to a Period-two Solution for a Class of Second Order Rational Difference Equations M. R. S. KulenoviC and 0. Merino
1. Introduction and Preliminaries
2. Main Results
2.1. Basin of attraction of prime period two solutions to
2.2. Basin of attraction of prime period two solutions to Iln--1 %+I = P+qYn+Yn-l
References
Two Normal Ordering Problems and Certain Sheffer Polynomials W. Lang
1. Introduction
2. Problem 1
3. Sheffer group and Jabotinsky subgroup
4. Problem 2
5. Alternative Approach t o Problem 2
6. Conclusion
Acknowledgements
References
Existence of Solutions for Even Ordered Boundary Value Problems on a Time Scale J. Henderson and B. A . Lawrence
1. Introduction
2. The Krasnosel’skiY-Zabreiko Fixed Point Theorem
3. Development of the Main Result
4. Existence Theorem
References
Asymptotics and Zeros of Symmetrically Coherent Pairs of Hermite Type M. G. De Bruin, W. G. M. Groenevelt, F. Marcelldn, H. G. Meijer and J. J. Moreno-Balcdzar
1. Introduction
2. Asymptotics
3. Zeros and its asymptotics
References
Asymptotic Boundary Value Problems for Discrete Systems M. Marini, S. Matucci and P. Rehdk
1. Introduction
2. The topological approach
3. Two asymptotic boundary value problems
The Limits of the Solutions of a Linear Delay Difference System H. Matsunaga
1. Introduction
2. Preliminaries
3. Characteristic equation
4. Main Result
References
Riemann-Hilbert Problem for a Generalized Nikishin System A . F. Moreno
1. Introduction
2. Generalized Nikishin systems
2.1. Orthogonality relations
2.2. The equilibrium problem
3. The Riemann-Hilbert problem
4. Normalization of the Riemann-Hilbert problem at infinity
References
Inequalities and Turhians for Some Special Functions A . Laforgia and P. Natalini
1. Introduction and backgrounds
2. TurAn-type inequalities
3. Proof of Theorem 1.1
4. Numerical results
References
Asymptotics in the Complex Plane of the Third Painlev6 Transcendent V. Yu. Novokshenov
1. Introduction
2. Direct monodromy problem
3. WKB approximation of *-function
4. Asymptotics at the turning points
5. Zeros of the elliptic function ansatz
References
Solutions of Dynamic Equations with Varying Time Scales B. A . Lawrence and R. W. Oberste-Vorth
1. Introduction
2. Families of Dynamic Equations on Time Scales
3. Convergence of Time Scales
4. Limits Over Time Scales
5. Convergence of Unique Solutions
6. Bifurcations over Time Scales
Acknowledgments
References
Totally Discrete and Eulerian Time Scales K . J. Hall and R. W. Oberste-Vorth
1. Introduction
2. Approximation in the Space of Time Scales
2.1. The Hausdorfl Metric
2.2. Approximation by Totally Discrete Times Scales
3. Parameterized Families of Dynamic Equations
3.1. A n Example: Parameter Space for Quadratic Equation
3.2. Review of the Dynamics of Quadratic Polynomials
3.3. Solutions over pZ+
3.4. Description of the Parameter Space
3.5. What Happens As p Tends To Q?
Acknowledgments
References
A Generalization of the Discrete UC Heirarchy and Its Reductions Y. Ohta
1. Introduction
2. Generalized Discrete UC Hierarchy
3. 2+1 Dimensional Integrable Systems
4. Single Bilinear Equation of Discrete UC Hierarchy
5. Concluding Remarks
Acknowledgments
References
The Dichotomy Character of z,+1 = PnXn+Ynxn-l with Period-two Coefficients C. H. Gibbons, S. Kalabuiic' and C. B. Overdeep
1. Introduction and Preliminaries
2. Character of Solutions
2.1. When go + T I 5 1 and 1 - q1 < TQ 5 1 (a subcase of Case 1 )
2.2. When r1 5 1 and ro = 91 + 1 (a Subcase of Cases 2 and 4 Taken Together)
2.3. When r1 5 90 + 1 and r g > q1 + 1 (Cases 3, 5, and 6 Taken Together)
3. Open Problems
Acknowledgement
References
Integral Comparison Theorems for Second Order Linear Dynamic Equations L. Erbe, A . Peterson and P. Reha'k
1. Preliminary Results
2. Main Results
References
Nonoscillations in Odd Order Difference Systems of Mixed Type S. Pinelas
1. Introduction
2. Nonoscillatory odd order systems
References
Superintegrability and Quasi-exact Solvability. The Anisotropic Oscillator G. S. Pogosyan
1. Introduction
2. Anisotropic oscillator
3. Conclusion
References
On the Ergodic and Special Properties of Generalized Boole Transformations A . K. Prykarpatsky and J. Feldman
1. Introduction
2. Invariant measures and ergodic transformations
3. Ergodic measures: an innert function approach
4. Invariant measures: the general case
Acknowledgements
References
Fock Representations for a Quadratic Commutation Relation C. Correia Ramos, N. Martins and J. Sousa Ramos
1. Introduction
2. The quadratic family
3. Types of Fock representations
4. Characterization of the set 23
5. Final remarks
Acknowledgments
References
Chaotic Discrete Learning Systems M. Merces Ramos and P. Sarreira
1. Introduction
2. Mathematical model
3. Topological entropy
4. Final considerations
Acknowledgements
References
On the Asymptotic Behavior of the Moments of Solutions of Stochastic Difference Equations J. Appleby, G. Berkolaiko and A. Rodkina
1. Introduction
2. Construction of convex estimate
3. Main result
4. Examples
Acknowledgments
References
Orthogonal Polynomials and the Bezout Identity A . Ronveaux, A . Zarzo, I. Area and E. Godoy
1. Introduction and Motivation
1 .l. Basic properties of classical orthogonal polynomials
1.2. Structure relations and derivative representations for classical families
1.2.1. Structure relations
1.2.2. Derivative representations
2. Recurrence relations between P,(z) and Bn-i(z), and between PA(z) and A,-z(z)
3. The three term recurrence relations for the &-family and the A,-family
4. Differential relations between A,(z) and B,(z)
5. About the orthogonality of families A,(z) and Bn(s)
6. Extensions: From continuous to discrete and q-discrete. Open problems
6.1. Extensions: Enlarging the familg P n ( x ) . Open problems
Acknowledgments
References
Information Entropy of Gegenbauer Polynomials J. I. de Vicente, S. Gandy and J. Scinchez-Ruiz
1. Introduction
2. Previous Results
3. Our Approach
3.1. Trigonometric Representations for Gegenbauer Polynomials
3.2. Evaluation of the Entropic Integral
3.3. Closed Form Expressions for the Entropy
Acknowledgments
References
Higher Genus Affine Lie Algebras of Krichever-Novikov Type M. Schlichenmaier
1. Introduction
2. The classical situation and some algebraic background
3. The higher genus case
4. Central extensions in higher genus
5. An example: The three-point genus zero case
6. An example: The torus case
References
Asymptotic Trichotomy of Solutions of a Class of Even Order Nonlinear Neutral Difference Equations with Quasidifferences E. Schmeidel
1. Introduction
2. Classification of nonoscillatory solutions
3. Sufficient conditions
References
Critical Groups for Iterated Maps C. Correia Ramos, N . Martins, R. Severino and J. Sousa Ramos
1. Introduction
2. Definitions and Notations
3. Main Theorem
Acknowledgments
References
Ideal Turbulence and Problems of Its Visualization A . N . Sharkovsky
1. Introduction
2. Spatial-temporal chaos in boundary value problems
3. Ideal turbulence: Defininions
4. Bifurcations leading to ideal turbulence
5. Visualization of ideal turbulence. Computer turbulence
References
Fine Structure of the Zeros of Orthogonal Polynomials: A Review B. Simon
1. Introduction
2. Prior Work
3. OPUC With Competing Exponential Decay
4. Clock Behavior Within the Nevai Class
5. Clock Behavior for Periodic OPUC
6. l/n Bounds
7. Zeros of Random POPUC
8. Zeros of Random OPUC
Acknowledgments
References
On the Symmetries of Integrable Partial Difference Equations A . Tongas
1. Introduction
2. Symmetries of quadrilateral equations
3. Symmetries of equation ( 5 )
4. Symmetries of the discrete potential KdV equation
5. Symmetry reduction on the lattice
6. Symmetry reductions to discrete Painlev6 equations
7. Concluding remarks
Acknowledgements
References
Heun Functions versus Elliptic Functions G. Valent
1. Introduction
2. From orthogonal polynomials to Heun functions
3. Non-generic solutions
3.1. Special hypergeometric cases
3.2. The "trivial" solution
3.3. Derivatives of Heun functions
3.4. Reduction to hypergeometric functions
4. Generic solutions
4.1. The I92 solutions of Heun's equation
4.2. An integral transform
4.3. Carlitz solutions
4.4. Second kind elliptic functions and Picard's theorem
4.4.1. Theorems for generic multipliers
4.4.2. Theorems for special multipliers
4.4.3. Picard's theorem
4.5. The rneromorphic solutions
4.6. Level one elliptic solutions
4.7. Finite-gap solutions
4.8. Finite-gap versus elliptic solutions
4.9. Conclusion
References
Discrete PainlevQ Equations for Recurrence Coefficients of Orthogonal Polynomials W. Van Assche
1. Introduction
2. F’reud weights
2.1. Generalized Herrnite polynomials
2.2. Freud weight r n = 4
2.3. Freud weight m = 6
2.4. bud's conjecture
3. Orthogonal polynomials on the unit circle
3.1. Modified Bessel polynomials
4. Discrete orthogonal polynomials
4.1. Charlier polynomials
4.2. Generalized Charlier polynomials
5. q-Orthogonal polynomials
5.1. Discrete q-Hermite I polynomials
5.2. Discrete q-Freud polynomials
5.3. Another discrete q-Freud case
Acknowledgments
References
Orthogonal Polynomials on R+ and Birth-Death Processes with Killing P. Coolen-Schrijner and E. A . van Doom
1. Introduction and main results
2. Preliminaries
3. Proof of Theorem 1.3 and related issues
4. Determinacy of the Smp and Hmp
5. Birth-death processes with killing
Acknowledgement
References
Computing Topological Invariants in Boundary Value Problems Reducible to Difference Equations R. Severino, A . Sharkovsky, J. Sousa Ramos and S. Vinagre
1. Introduction
2. Preliminaries
3. Symbolic dynamics
4. Results
References
Abel’s Method on Summation by Parts and Bilateral Well-poised yJ~s-series Identities W. C. Chu
1. Bilateral Well-Poised s?,!~a-Series Identities
2. New Proofs via Abel’s Method on Summation by Parts
3. q-Analogue of Dixon’s Theorem
Acknowledgements
References
(2 + 1)-dimensional Lattice Hierarchies Derived from Discrete Operator Zero Curvature Equation 2.-N. Zhu
1. Introduction
2. 2+1 dimensional lattice systems derived from discrete operator zero curvature equation
Acknowledgments
References