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Fibonacci and Lucas Numbers with Applications, Volume II

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$100.99

Fibonacci and Lucas Numbers with Applications, Volume II

Thomas Koshy

ISBN: 978-1-118-74218-1 December 2018 752 Pages

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Description

Volume II provides an advanced approach to the extended gibonacci family, which includes Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas, Vieta, Vieta-Lucas, and Chebyshev polynomials of both kinds. This volume offers a uniquely unified, extensive, and historical approach that will appeal to both students and professional mathematicians. 

As in Volume I, Volume II focuses on problem-solving techniques such as pattern recognition; conjecturing; proof-techniques, and applications. It offers a wealth of delightful opportunities to explore and experiment, as well as plentiful material for group discussions, seminars, presentations, and collaboration.

In addition, the material covered in this book promotes intellectual curiosity, creativity, and ingenuity.

Volume II features: 

  • A wealth of examples, applications, and exercises of varying degrees of difficulty and sophistication.
  • Numerous combinatorial and graph-theoretic proofs and techniques.
  • A uniquely thorough discussion of gibonacci subfamilies, and the fascinating relationships that link them.
  • Examples of the beauty, power, and ubiquity of the extended gibonacci family.
  • An introduction to tribonacci polynomials and numbers, and their combinatorial and graph-theoretic models.
  • Abbreviated solutions provided for all odd-numbered exercises.
  • Extensive references for further study.

This volume will be a valuable resource for upper-level undergraduates and graduate students, as well as for independent study projects, undergraduate and graduate theses. It is the most comprehensive work available, a welcome addition for gibonacci enthusiasts in computer science, electrical engineering, and physics, as well as for creative and curious amateurs.

List of Symbols xiii

Preface xv

31. Fibonacci and Lucas Polynomials I 1

31.1. Fibonacci and Lucas Polynomials 3

31.2. Pascal’s Triangle 18

31.3. Additional Explicit Formulas 22

31.4. Ends of the Numbers ln 25

31.5. Generating Functions 26

31.6. Pell and Pell–Lucas Polynomials 27

31.7. Composition of Lucas Polynomials 33

31.8. De Moivre-like Formulas 35

31.9. Fibonacci–Lucas Bridges 36

31.10. Applications of Identity (31.51) 37

31.11. Infinite Products 48

31.12. Putnam Delight Revisited 51

31.13. Infinite Simple Continued Fraction 54

32. Fibonacci and Lucas Polynomials II 65

32.1. Q-Matrix 65

32.2. Summation Formulas 67

32.3. Addition Formulas 71

32.4. A Recurrence for n2 76

32.5. Divisibility Properties 82

33. Combinatorial Models II 87

33.1. A Model for Fibonacci Polynomials 87

33.2. Breakability 99

33.3. A Ladder Model 101

33.4. A Model for Pell–Lucas Polynomials: Linear Boards 102

33.5. Colored Tilings 103

33.6. A New Tiling Scheme 104

33.7. A Model for Pell–Lucas Polynomials: Circular Boards 107

33.8. A Domino Model for Fibonacci Polynomials 114

33.9. Another Model for Fibonacci Polynomials 118

34. Graph-Theoretic Models II 125

34.1. Q-Matrix and Connected Graph 125

34.2. Weighted Paths 126

34.3. Q-Matrix Revisited 127

34.4. Byproducts of the Model 128

34.5. A Bijection Algorithm 136

34.6. Fibonacci and Lucas Sums 137

34.7. Fibonacci Walks 140

35. Gibonacci Polynomials 145

35.1. Gibonacci Polynomials 145

35.2. Differences of Gibonacci Products 159

35.3. Generalized Lucas and Ginsburg Identities 174

35.4. Gibonacci and Geometry 181

35.5. Additional Recurrences 184

35.6. Pythagorean Triples 188

36. Gibonacci Sums 195

36.1. Gibonacci Sums 195

36.2. Weighted Sums 206

36.3. Exponential Generating Functions 209

36.4. Infinite Gibonacci Sums 215

37. Additional Gibonacci Delights 233

37.1. Some Fundamental Identities Revisited 233

37.2. Lucas and Ginsburg Identities Revisited 238

37.3. Fibonomial Coefficients 247

37.4. Gibonomial Coefficients 250

37.5. Additional Identities 260

37.6. Strazdins’ Identity 264

38. Fibonacci and Lucas Polynomials III 269

38.1. Seiffert’s Formulas 270

38.2. Additional Formulas 294

38.3. Legendre Polynomials 314

39. Gibonacci Determinants 321

39.1. A Circulant Determinant 321

39.2. A Hybrid Determinant 323

39.3. Basin’s Determinant 333

39.4. Lower Hessenberg Matrices 339

39.5. Determinant with a Prescribed First Row 343

40. Fibonometry II 347

40.1. Fibonometric Results 347

40.2. Hyperbolic Functions 356

40.3. Inverse Hyperbolic Summation Formulas 361

41. Chebyshev Polynomials 371

41.1. Chebyshev Polynomials Tn(x) 372

41.2. Tn(x) and Trigonometry 384

41.3. Hidden Treasures in Table 41.1 386

41.4. Chebyshev Polynomials Un(x) 396

41.5. Pell’s Equation 398

41.6. Un(x) and Trigonometry 399

41.7. Addition and Cassini-like Formulas 401

41.8. Hidden Treasures in Table 41.8 402

41.9. A Chebyshev Bridge 404

41.10. Tn and Un as Products 405

41.11. Generating Functions 410

42. Chebyshev Tilings 415

42.1. Combinatorial Models for Un 415

42.2. Combinatorial Models for Tn 420

42.3. Circular Tilings 425

43. Bivariate Gibonacci Family I 429

43.1. Bivariate Gibonacci Polynomials 429

43.2. Bivariate Fibonacci and Lucas Identities 430

43.3. Candido’s Identity Revisited 439

44. Jacobsthal Family 443

44.1. Jacobsthal Family 444

44.2. Jacobsthal Occurrences 450

44.3. Jacobsthal Compositions 452

44.4. Triangular Numbers in the Family 459

44.5. Formal Languages 468

44.6. A USA Olympiad Delight 480

44.7. A Story of 1, 2, 7, 42, 429,…483

44.8. Convolutions 490

45. Jacobsthal Tilings and Graphs 499

45.1. 1 × n Tilings 499

45.2. 2 × n Tilings 505

45.3. 2 × n Tubular Tilings 510

45.4. 3 × n Tilings 514

45.5. Graph-Theoretic Models 518

45.6. Digraph Models 522

46. Bivariate Tiling Models 537

46.1. A Model for 𝑓n(x, y) 537

46.2. Breakability 539

46.3. Colored Tilings 542

46.4. A Model for ln(x, y) 543

46.5. Colored Tilings Revisited 545

46.6. Circular Tilings Again 547

47. Vieta Polynomials 553

47.1. Vieta Polynomials 554

47.2. Aurifeuille’s Identity 567

47.3. Vieta–Chebyshev Bridges 572

47.4. Jacobsthal–Chebyshev Links 573

47.5. Two Charming Vieta Identities 574

47.6. Tiling Models for Vn 576

47.7. Tiling Models for 𝑣n(x) 582

48. Bivariate Gibonacci Family II 591

48.1. Bivariate Identities 591

48.2. Additional Bivariate Identities 594

48.3. A Bivariate Lucas Counterpart 599

48.4. A Summation Formula for 𝑓2n(x, y) 600

48.5. A Summation Formula for l2n(x, y) 602

48.6. Bivariate Fibonacci Links 603

48.7. Bivariate Lucas Links 606

49. Tribonacci Polynomials 611

49.1. Tribonacci Numbers 611

49.2. Compositions with Summands 1, 2, and 3 613

49.3. Tribonacci Polynomials 616

49.4. A Combinatorial Model 618

49.5. Tribonacci Polynomials and the Q-Matrix 624

49.6. Tribonacci Walks 625

49.7. A Bijection between the Two Models 627

Appendix 631

A.1. The First 100 Fibonacci and Lucas Numbers 631

A.2. The First 100 Pell and Pell–Lucas Numbers 634

A.3. The First 100 Jacobsthal and Jacobsthal–Lucas Numbers 638

A.4. The First 100 Tribonacci Numbers 642

Abbreviations 644

Bibliography 645

Solutions to Odd-Numbered Exercises 661

Index 725