Difference Equations, Special Functions and Orthogonal Polynomials: Proceedings of the International Conference: Munich, Germany 25-30, July 2005

Contents......Page 10
Preface......Page 8
1. Introduction......Page 16
2. Relevant properties......Page 18
2.1. Connections with some known matrices......Page 20
3.1. Bernstein and Legendre polynomials......Page 22
3.2. Bernoulli and Euler polynomials......Page 24
3.3. Hermite polynomials......Page 26
4. Matrices, difference equations and polynomials......Page 27
References......Page 30
1. Introduction......Page 32
2. Proof of Theorem 1.3......Page 34
References......Page 37
1. Introduction......Page 38
2. Properties of the Solution of (1)-(3)......Page 40
3.1. Construction of the difference scheme......Page 42
4.1. Stability Bounds......Page 46
5. Algorithm and Numerical Results......Page 51
References......Page 54
1. Introduction......Page 56
2. Preliminary results for p = 3......Page 57
3. The particular case f ( z , y ) = u ( z ) p ( y )......Page 62
Bibliography......Page 64
1. Introduction and results......Page 66
2. Proof of the main results......Page 70
3. Stieltjes moment problems......Page 73
4. Examples......Page 74
5. The logarithmic growth scale......Page 75
6. Appendix: The Phragmh-Lindeliif indicator of some functions of order zero......Page 88
References......Page 92
1. Introduction......Page 95
2. A sequence of trigonometric polynomials......Page 96
3. Favard’s type theorem......Page 102
References......Page 104
1. Introduction......Page 105
2.1. A general result......Page 106
2.2. Shener polynomials......Page 107
2.3. Generalized Hennite polynomials......Page 108
2.5. Orthogonal Shefler polynomials......Page 109
3.1. General expression......Page 110
3.2. Application to Gould-Hopper polynomials......Page 112
References......Page 113
1. Introduction and preliminary results......Page 115
2. Proof of the main result......Page 117
3. Special cases......Page 118
3.3. Case d 21.......Page 119
4.1. A ( d + 1)-order recurrence relation......Page 120
4.2. d -dimensional finctional......Page 121
5. Chebyshev d-OPS of the first kind......Page 123
References......Page 125
1. Introduction......Page 127
2. The case of (0, 1, . . . , T , T + m) interpolation......Page 128
3. The case of (O,m, m + 1 , . . . , m + T ) interpolation......Page 130
4. Proofs for Sec. 2......Page 131
5. Proofs for Sec. 3......Page 135
References......Page 136
1. Introduction......Page 137
2. Some definitions and preliminary results......Page 138
3. Construction of symmetric semiclassical linear functionals of class 2......Page 140
References......Page 145
1. Introduction......Page 146
2. Preliminaries......Page 150
3. Escape Sierpinski Curve Julia Sets......Page 151
4. Buried Sierpinski Curves......Page 155
5. Structurally Unstable Sierpinski Curves......Page 157
6. Final Comments and Conjectures......Page 161
References......Page 163
1. Introduction......Page 164
1.2. Preliminary......Page 165
1.2.3. Points of strict egress and their geometrical sense......Page 166
2. Main Result......Page 167
2.1.1. Specification of the general scheme of the proof......Page 168
2.1.2. Auxiliary mapping R1......Page 169
2.1.3. Auxiliary mapping R:2......Page 172
References......Page 173
1. Introduction......Page 174
2. Main results......Page 175
3. Discrete versus continuous case......Page 180
References......Page 182
1. Introduction......Page 183
2. Main results......Page 185
3. Symplectic factorizations and index results......Page 188
References......Page 192
A Renaissance for a q-umbra1 Calculus T. Ernst......Page 193
References......Page 202
1. Introduction......Page 204
3. The fourth-order differential equation......Page 205
4. Higher-order differential equations......Page 210
5. The fourth-order differential expression LM......Page 211
6. Hilbert function spaces......Page 212
8. Differential operators in L2((0, 00); z)......Page 213
10. Boundary properties at O+......Page 214
11. Explicit boundary condition functions at O+......Page 216
12. Spectral properties of the fourth-order Bessel-type operators......Page 217
14. Self-adjoint operator s k in L2([0, 00); m k )......Page 218
16. Distributional orthogonality relationships......Page 219
17. The generalised Hankel transform......Page 220
18. The Plum partial differential equation......Page 224
References......Page 226
1. Introduction......Page 228
2. Discrete Laplacian and conductance......Page 232
3. Systoles in discrete dynamical systems......Page 236
References......Page 237
1. Introduction......Page 239
2. Rational and exponential solutions of linear differential equations......Page 241
2.1. LDE with coemcients in C ( x )......Page 242
2.2. LDE with coeficients in exponential extensions......Page 244
3. Liouvillian solutions of linear differential equations......Page 246
4. Solutions of linear differential equations in term of special functions......Page 248
5. Linear differential systems......Page 249
6. Linear difference equations......Page 252
References......Page 253
2. Preliminary......Page 256
3.1. Linear case......Page 257
3.2. Nonlinear case......Page 260
References......Page 263
1. Introduction......Page 264
2. Invariants and vector fields......Page 265
3. Computation of the characteristic algebra......Page 267
4. Characteristic algebra for the discrete Liouville equation......Page 269
5. How to find the invariants?......Page 271
References......Page 272
1 Introduction......Page 273
2 Global Attractivity of Eq.(l) and Others......Page 274
References......Page 280
1. Introduction and Motivation......Page 281
2. Main Results - Nonnegativity......Page 284
3. Positive Definiteness......Page 287
4. Continuous-Time Case......Page 288
References......Page 289
1. Motivation......Page 291
3. Discrete-time problems......Page 293
3.1. Time discretization of with Runge-Kutta methods......Page 294
4. Discrete monotonicity of RK methods......Page 295
4.1. Discrete monotonicity of the explicit Euler method......Page 296
References......Page 299
1. Introduction and preliminaries......Page 301
2. Successive approximations......Page 303
3. Convergence......Page 305
References......Page 308
1. Introduction......Page 309
2. The model and some considerations about triangular maps......Page 311
3. Chaotic behavior in the map F......Page 313
References......Page 318
1. Introduction......Page 320
2. The discrete perturbation technique......Page 321
3. Reduction of the lattice sine-Gordon equation......Page 325
Acknowledgments......Page 328
References......Page 329
1. Equation k ( t ) = a z ( t - 7 ) V8 Equation Zn - Zn-1 = UZn-k Euler - Levin and May (1976)......Page 330
2. System &(t) = A s ( t - T ) US System xn - xn-1 = A5n-k Rekhlitskii (1956) - Levitskaya (2005)......Page 331
3.1. Preliminaries......Page 333
3.2. Results......Page 334
3.4. Summary......Page 335
References......Page 339
1. Demonstrations with Computer Algebra......Page 340
2. Classical Orthogonal Polynomials......Page 342
3. Hypergeometric Functions......Page 344
4. Computation of the Recurrence Coefficients......Page 346
5. Zeilberger’s Algorithm......Page 350
6. Petkovsek-van Hoeij Algorithm......Page 354
7. Recurrence Operators......Page 355
8. Classical Orthogonal Polynomial Solutions of Recurrence Equations......Page 356
References......Page 358
1. Introduction and Preliminaries......Page 359
2.1. Basin of attraction of prime period two solutions to......Page 363
2.2. Basin of attraction of prime period two solutions to Iln--1 %+I = P+qYn+Yn-l......Page 366
References......Page 368
1. Introduction......Page 369
2. Problem 1......Page 370
3. Sheffer group and Jabotinsky subgroup......Page 373
4. Problem 2......Page 374
5. Alternative Approach t o Problem 2......Page 376
Acknowledgements......Page 377
References......Page 378
1. Introduction......Page 384
2. The Krasnosel’skiY-Zabreiko Fixed Point Theorem......Page 386
3. Development of the Main Result......Page 389
4. Existence Theorem......Page 391
References......Page 392
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......Page 393
1. Introduction......Page 394
2. Asymptotics......Page 395
3. Zeros and its asymptotics......Page 402
References......Page 408
1. Introduction......Page 409
2. The topological approach......Page 410
3. Two asymptotic boundary value problems......Page 413
1. Introduction......Page 419
2. Preliminaries......Page 420
3. Characteristic equation......Page 422
4. Main Result......Page 423
References......Page 426
1. Introduction......Page 427
2. Generalized Nikishin systems......Page 428
2.1. Orthogonality relations......Page 429
2.2. The equilibrium problem......Page 430
3. The Riemann-Hilbert problem......Page 431
4. Normalization of the Riemann-Hilbert problem at infinity......Page 432
References......Page 436
1. Introduction and backgrounds......Page 437
2. TurAn-type inequalities......Page 440
3. Proof of Theorem 1.1......Page 442
4. Numerical results......Page 444
References......Page 445
1. Introduction......Page 447
2. Direct monodromy problem......Page 452
3. WKB approximation of *-function......Page 454
4. Asymptotics at the turning points......Page 458
5. Zeros of the elliptic function ansatz......Page 463
References......Page 466
1. Introduction......Page 467
2. Families of Dynamic Equations on Time Scales......Page 468
3. Convergence of Time Scales......Page 469
4. Limits Over Time Scales......Page 470
5. Convergence of Unique Solutions......Page 471
6. Bifurcations over Time Scales......Page 474
References......Page 475
1. Introduction......Page 477
2.1. The Hausdorfl Metric......Page 478
2.2. Approximation by Totally Discrete Times Scales......Page 479
3. Parameterized Families of Dynamic Equations......Page 480
3.1. A n Example: Parameter Space for Quadratic Equation......Page 481
3.2. Review of the Dynamics of Quadratic Polynomials......Page 482
3.3. Solutions over pZ+......Page 483
References......Page 484
1. Introduction......Page 486
2. Generalized Discrete UC Hierarchy......Page 487
3. 2+1 Dimensional Integrable Systems......Page 490
4. Single Bilinear Equation of Discrete UC Hierarchy......Page 493
References......Page 494
1. Introduction and Preliminaries......Page 495
2.1. When go + T I 5 1 and 1 - q1 < TQ 5 1 (a subcase of Case 1 )......Page 501
2.2. When r1 5 1 and ro = 91 + 1 (a Subcase of Cases 2 and 4 Taken Together)......Page 503
2.3. When r1 5 90 + 1 and r g > q1 + 1 (Cases 3, 5, and 6 Taken Together)......Page 504
3. Open Problems......Page 506
Acknowledgement......Page 508
References......Page 509
1. Preliminary Results......Page 512
2. Main Results......Page 514
References......Page 521
1. Introduction......Page 522
2. Nonoscillatory odd order systems......Page 527
References......Page 534
1. Introduction......Page 535
2. Anisotropic oscillator......Page 536
3. Conclusion......Page 540
References......Page 541
1. Introduction......Page 542
2. Invariant measures and ergodic transformations......Page 543
3. Ergodic measures: an innert function approach......Page 545
4. Invariant measures: the general case......Page 548
References......Page 551
1. Introduction......Page 552
2. The quadratic family......Page 553
3. Types of Fock representations......Page 555
4. Characterization of the set 23......Page 557
5. Final remarks......Page 559
Acknowledgments......Page 560
References......Page 561
1. Introduction......Page 562
2. Mathematical model......Page 564
3. Topological entropy......Page 565
References......Page 570
On the Asymptotic Behavior of the Moments of Solutions of Stochastic Difference Equations J. Appleby, G. Berkolaiko and A. Rodkina......Page 572
2. Construction of convex estimate......Page 573
3. Main result......Page 574
Acknowledgments......Page 579
References......Page 580
Orthogonal Polynomials and the Bezout Identity A . Ronveaux, A . Zarzo, I. Area and E. Godoy......Page 581
1. Introduction and Motivation......Page 582
1 .l. Basic properties of classical orthogonal polynomials......Page 583
1.2. Structure relations and derivative representations for classical families......Page 584
1.2.2. Derivative representations......Page 585
2. Recurrence relations between P,(z) and Bn-i(z), and between PA(z) and A,-z(z)......Page 586
3. The three term recurrence relations for the &-family and the A,-family......Page 587
4. Differential relations between A,(z) and B,(z)......Page 588
5. About the orthogonality of families A,(z) and Bn(s)......Page 589
6. Extensions: From continuous to discrete and q-discrete. Open problems......Page 590
6.1. Extensions: Enlarging the familg P n ( x ) . Open problems......Page 591
References......Page 593
1. Introduction......Page 594
2. Previous Results......Page 596
3.1. Trigonometric Representations for Gegenbauer Polynomials......Page 597
3.2. Evaluation of the Entropic Integral......Page 598
3.3. Closed Form Expressions for the Entropy......Page 600
References......Page 602
1. Introduction......Page 604
2. The classical situation and some algebraic background......Page 605
3. The higher genus case......Page 607
4. Central extensions in higher genus......Page 609
5. An example: The three-point genus zero case......Page 611
6. An example: The torus case......Page 613
References......Page 614
1. Introduction......Page 615
2. Classification of nonoscillatory solutions......Page 617
3. Sufficient conditions......Page 620
References......Page 624
1. Introduction......Page 625
2. Definitions and Notations......Page 626
3. Main Theorem......Page 629
References......Page 630
1. Introduction......Page 632
2. Spatial-temporal chaos in boundary value problems......Page 633
3. Ideal turbulence: Defininions......Page 637
4. Bifurcations leading to ideal turbulence......Page 641
5. Visualization of ideal turbulence. Computer turbulence......Page 645
References......Page 648
1. Introduction......Page 651
2. Prior Work......Page 655
3. OPUC With Competing Exponential Decay......Page 657
4. Clock Behavior Within the Nevai Class......Page 658
5. Clock Behavior for Periodic OPUC......Page 660
7. Zeros of Random POPUC......Page 661
8. Zeros of Random OPUC......Page 663
References......Page 665
1. Introduction......Page 669
2. Symmetries of quadrilateral equations......Page 670
3. Symmetries of equation ( 5 )......Page 673
4. Symmetries of the discrete potential KdV equation......Page 674
5. Symmetry reduction on the lattice......Page 675
6. Symmetry reductions to discrete Painlev6 equations......Page 676
Acknowledgements......Page 677
References......Page 678
1. Introduction......Page 679
2. From orthogonal polynomials to Heun functions......Page 680
3.1. Special hypergeometric cases......Page 682
3.3. Derivatives of Heun functions......Page 683
3.4. Reduction to hypergeometric functions......Page 684
4.1. The I92 solutions of Heun's equation......Page 685
4.2. An integral transform......Page 686
4.3. Carlitz solutions......Page 687
4.4. Second kind elliptic functions and Picard's theorem......Page 688
4.4.1. Theorems for generic multipliers......Page 690
4.4.2. Theorems for special multipliers......Page 691
4.4.3. Picard's theorem......Page 692
4.5. The rneromorphic solutions......Page 694
4.6. Level one elliptic solutions......Page 695
4.7. Finite-gap solutions......Page 697
4.8. Finite-gap versus elliptic solutions......Page 699
References......Page 700
1. Introduction......Page 702
2.1. Generalized Herrnite polynomials......Page 705
2.2. Freud weight r n = 4......Page 707
2.3. Freud weight m = 6......Page 711
3. Orthogonal polynomials on the unit circle......Page 713
3.1. Modified Bessel polynomials......Page 714
4. Discrete orthogonal polynomials......Page 717
4.1. Charlier polynomials......Page 718
4.2. Generalized Charlier polynomials......Page 719
5. q-Orthogonal polynomials......Page 725
5.1. Discrete q-Hermite I polynomials......Page 726
5.2. Discrete q-Freud polynomials......Page 728
5.3. Another discrete q-Freud case......Page 732
Acknowledgments......Page 734
References......Page 739
1. Introduction and main results......Page 741
2. Preliminaries......Page 745
3. Proof of Theorem 1.3 and related issues......Page 748
4. Determinacy of the Smp and Hmp......Page 751
5. Birth-death processes with killing......Page 752
References......Page 754
1. Introduction......Page 756
2. Preliminaries......Page 757
3. Symbolic dynamics......Page 760
4. Results......Page 763
References......Page 766
Abel’s Method on Summation by Parts and Bilateral Well-poised yJ~s-series Identities W. C. Chu......Page 767
1. Bilateral Well-Poised s?,!~a-Series Identities......Page 769
2. New Proofs via Abel’s Method on Summation by Parts......Page 771
3. q-Analogue of Dixon’s Theorem......Page 774
Acknowledgements......Page 775
References......Page 776
1. Introduction......Page 777
2. 2+1 dimensional lattice systems derived from discrete operator zero curvature equation......Page 778
Acknowledgments......Page 787
References......Page 788