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.
Table of Contents
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
