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…mehr
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 andsophistication. * 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 andgraph-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.
Thomas Koshy, PhD, is the author of eleven books and numerous articles. As a professor of Mathematics at Framingham State University in Framingham, Massachusetts, he received the Distinguished Service Award, Citation for Meritorious Service, Commonwealth Citation for Outstanding Performance, as well as Faculty of the Year. He received his PhD in Algebraic Coding Theory from Boston University, under the guidance of Dr. Edwin Weiss. "Dr. Koshy is a meticulous researcher who shares his encyclopedic knowledge regarding Fibonacci and Lucas numbers in Fibonacci and Lucas Numbers, Volume I. In Volume II, he extends all of those wonderful ideas and identities to the Gibonacci polynomials, the "grandfathers" of the Fibonacci and Lucas polynomials. Writing in a readable style and including many examples and exercises, Koshy ties together Fibonacci and Lucas polynomials with Chebyshev, Jacobsthal, and Vieta polynomials. Once again, Koshy has compiled lore from diverse sources into one understandable and intriguing volume." Marjorie Bicknell Johnson
Inhaltsangabe
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 n(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 n(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 2n(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
