Fibonacci and Lucas numbers with applications

著者

• Koshy, Thomas

書誌事項

Fibonacci and Lucas numbers with applications

Thomas Koshy

（Pure and applied mathematics）

Wiley, c2018-2019

2nd ed

• v. 1
• v. 2

注記

Paging, v. 1: xx, 680 p. -- v. 2: xviii, 729 p.

Includes bibliographical references (v. 1: p. 599-617., v. 2: p. 645-660) and index

"This edition first published 2019" -- T.p. verso, v. 2

"Description: Second edition. ..." -- LCC data, V. 2

内容説明・目次

巻冊次

v. 2 ISBN 9781118742082

内容説明

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.

目次

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 x n Tilings 499 45.2. 2 x n Tilings 505 45.3. 2 x n Tubular Tilings 510 45.4. 3 x 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

巻冊次

v. 1 ISBN 9781118742129

内容説明

Praise for the First Edition " ...beautiful and well worth the reading ... with many exercises and a good bibliography, this book will fascinate both students and teachers." Mathematics Teacher Fibonacci and Lucas Numbers with Applications, Volume I, Second Edition provides a user-friendly and historical approach to the many fascinating properties of Fibonacci and Lucas numbers, which have intrigued amateurs and professionals for centuries. Offering an in-depth study of the topic, this book includes exciting applications that provide many opportunities to explore and experiment. In addition, the book includes a historical survey of the development of Fibonacci and Lucas numbers, with biographical sketches of important figures in the field. Each chapter features a wealth of examples, as well as numeric and theoretical exercises that avoid using extensive and time-consuming proofs of theorems. The Second Edition offers new opportunities to illustrate and expand on various problem-solving skills and techniques. In addition, the book features: * A clear, comprehensive introduction to one of the most fascinating topics in mathematics, including links to graph theory, matrices, geometry, the stock market, and the Golden Ratio * Abundant examples, exercises, and properties throughout, with a wide range of difficulty and sophistication * Numeric puzzles based on Fibonacci numbers, as well as popular geometric paradoxes, and a glossary of symbols and fundamental properties from the theory of numbers * A wide range of applications in many disciplines, including architecture, biology, chemistry, electrical engineering, physics, physiology, and neurophysiology The Second Edition is appropriate for upper-undergraduate and graduate-level courses on the history of mathematics, combinatorics, and number theory. The book is also a valuable resource for undergraduate research courses, independent study projects, and senior/graduate theses, as well as a useful resource for computer scientists, physicists, biologists, and electrical engineers. Thomas Koshy, PhD, is Professor Emeritus of Mathematics at Framingham State University in Massachusetts and author of several books and numerous articles on mathematics. His work has been recognized by the Association of American Publishers, and he has received many awards, including the Distinguished Faculty of the Year. Dr. Koshy received his PhD in Algebraic Coding Theory from Boston University. "Anyone who loves mathematical puzzles, number theory, and Fibonacci numbers will treasure this book. Dr. Koshy has compiled Fibonacci lore from diverse sources into one understandable and intriguing volume, [interweaving] a historical flavor into an array of applications." Marjorie Bicknell-Johnson

目次

1 Leonardo Fibonacci 9 2 Fibonacci Numbers 13 2.1 Fibonacci's Rabbits 13 2.2 Fibonacci Numbers 14 2.3 Fibonacci and Lucas Curiosities 17 3 Fibonacci Numbers in Nature 27 3.1 Fibonacci, Flowers, and Trees 28 3.2 Fibonacci and Male Bees 31 3.3 Fibonacci, Lucas, and Subsets 32 3.4 Fibonacci and Sewage Treatment 34 3.5 Fibonacci and Atoms 35 3.6 Fibonacci and Reflections 36 3.7 Paraffins and Cycloparaffins 38 3.8 Fibonacci and Music 41 3.9 Fibonacci and Poetry 42 3.10 Fibonacci and Neurophysiology 43 3.11 Electrical Networks 45 4 Additional Fibonacci and Lucas Occurrences 53 4.1 Fibonacci Occurrences 53 4.2 Fibonacci and Compositions 58 4.3 Fibonacci and Permutations 61 4.4 Fibonacci and Generating Sets 63 4.5 Fibonacci and Graph Theory 64 4.6 Fibonacci Walks 66 4.7 Fibonacci Trees 68 4.8 Partitions 71 4.9 Fibonacci and the Stock Market 72 5 Fibonacci and Lucas Identities 77 5.1 Spanning Tree of a Connected Graph 79 5.2 Binet's Formulas 83 5.3 Cyclic Permutations and Lucas Numbers 91 5.4 Compositions Revisited 94 5.5 Number of Digits in Fn and Ln 94 5.6 Theorem 5.8 Revisited 95 5.7 Catalan's Identity 99 5.8 Additional Fibonacci and Lucas Identities 102 5.9 Fermat and Fibonacci 108 5.10 Fibonacci and 110 6 Geometric Illustrations and Paradoxes 117 6.1 Geometric Illustrations 117 6.2 Candido's Identity 121 6.3 Fibonacci Tessellations 123 6.4 Lucas Tessellations 123 6.5 Geometric Paradoxes 124 6.6 Cassini-Based Paradoxes 124 6.7 Additional Paradoxes 129 7 Gibonacci Numbers 133 7.1 Gibonacci Numbers 133 7.2 Germain's Identity 139 8 Additional Fibonacci and Lucas Formulas 145 8.1 New Explicit Formulas 145 8.2 Additional Formulas 148 9 The Euclidean Algorithm 159 9.1 The Euclidean Algorithm 160 9.2 Formula (5.5) Revisited 162 9.3 Lame's Theorem 164 10 Divisibility Properties 167 10.1 Fibonacci Divisibility 167 10.2 Lucas Divisibility 173 10.3 Fibonacci and Lucas Ratios 173 10.4 An Altered Fibonacci Sequence 178 11 Pascal's Triangle 185 11.1 Binomial Coefficients 185 11.2 Pascal's Triangle 186 11.3 Fibonacci Numbers and Pascal's Triangle 188 11.4 Another Explicit Formula for Ln 191 11.5 Catalan's Formula 192 11.6 Additional Identities 192 11.7 Fibonacci Paths of a Rook on a Chessboard 194 12 Pascal-like Triangles 199 12.1 Sums of Like-Powers 199 12.2 An Alternate Formula for Ln 202 12.3 Differences of Like-Powers 202 12.4 Catalan's Formula Revisited 204 12.5 A Lucas Triangle 205 12.6 Powers of Lucas Numbers 209 12.7 Variants of Pascal's Triangle 211 13 Recurrences and Generating Functions 219 13.1 LHRWCCs 219 13.2 Generating Functions 223 13.3 A Generating Function For F3n 233 13.4 A Generating Function For F3 n 234 13.5 Summation Formula (5.1) Revisited 234 13.6 A List of Generating Functions 235 13.7 Compositions Revisited 238 13.8 Exponential Generating Functions 239 13.9 Hybrid Identities 241 13.10Identities Using the Differential Operator 242 14 Combinatorial Models I 249 14.1 A Fibonacci Tiling Model 249 14.2 A Circular Tiling Model 255 14.3 Path Graphs Revisited 259 14.4 Cycle Graphs Revisited 262 14.5 Tadpole Graphs 263 15 Hosoya's Triangle 271 15.1 Recursive Definition 271 15.2 A Magic Rhombus 273 16 The Golden Ratio 279 16.1 Ratios of Consecutive Fibonacci Numbers 279 16.2 The Golden Ratio 281 16.3 Golden Ratio as Nested Radicals 285 16.4 Newton's Approximation Method 286 16.5 The Ubiquitous Golden Ratio 288 16.6 Human Body and the Golden Ratio 289 16.7 Violin and the Golden Ratio 290 16.8 Ancient Floor Mosaics and the Golden Ratio 290 16.9 Golden Ratio in an Electrical Network 290 16.10Golden Ratio in Electrostatics 291 16.11Golden Ratio by Origami 292 16.12Differential Equations 297 16.13Golden Ratio in Algebra 299 16.14Golden Ratio in Geometry 300 17 Golden Triangles and Rectangles 309 17.1 Golden Triangle 309 17.2 Golden Rectangles 314 17.3 The Parthenon 317 17.4 Human Body and the Golden Rectangle 318 17.5 Golden Rectangle and the Clock 319 17.6 Straightedge and Compass Construction 320 17.7 Reciprocal of a Rectangle 321 17.8 Logarithmic Spiral 322 17.9 Golden Rectangle Revisited 324 17.10Supergolden Rectangle 324 18 Figeometry 329 18.1 The Golden Ratio and Plane Geometry 329 18.2 The Cross of Lorraine 335 18.3 Fibonacci Meets Appollonius 337 18.4 A Fibonacci Spiral 338 18.5 Regular Pentagons 339 18.6 Trigonometric Formulas for Fn 343 18.7 Regular Decagon 347 18.8 Fifth Roots of Unity 348 18.9 A Pentagonal Arch 351 18.10 Regular Icosahedron and Dodecahedron 351 18.11 Golden Ellipse 352 18.12 Golden Hyperbola 354 19 Continued Fractions 361 19.1 Finite Continued Fractions 361 19.2 Convergents of a Continued Fraction 364 19.3 Infinite Continued Fractions 366 19.4 A Nonlinear Diophantine Equation 368 20 Fibonacci Matrices 371 20.1 The Q-Matrix 371 20.2 Eigenvalues of Qn 378 20.3 Fibonacci and Lucas Vectors 384 20.4 An Intriguing Fibonacci Matrix 386 20.5 An Infinite-Dimensional Lucas Matrix 391 20.6 An Infinite-Dimensional Gibonacci Matrix 397 20.7 The Lambda Function 398 21 Graph-theoretic Models I 407 21.1 A Graph-theoretic Model for Fibonacci Numbers 407 21.2 Byproducts of the Combinatorial Models 409 21.3 Summation Formulas 415 22 Fibonacci Determinants 419 22.1 An Application to Graph Theory 419 22.2 The Singularity of Fibonacci Matrices 425 22.3 Fibonacci and Analytic Geometry 427 23 Fibonacci and Lucas Congruences 437 23.1 Fibonacci Numbers Ending in Zero 437 23.2 Lucas Numbers Ending in Zero 437 23.3 Additional Congruences 438 23.4 Lucas Squares 439 23.5 Fibonacci Squares 440 23.6 A Generalized Fibonacci Congruence 442 23.7 Fibonacci and Lucas Periodicities 449 23.8 Lucas Squares Revisited 450 23.9 Periodicities Modulo 10n 452 24 Fibonacci and Lucas Series 461 24.1 A Fibonacci Series 461 24.2 A Lucas Series 463 24.3 Fibonacci and Lucas Series Revisited 464 24.4 A Fibonacci Power Series 467 24.5 Gibonacci Series 472 24.6 Additional Fibonacci Series 474 25 Weighted Fibonacci and Lucas Sums 481 25.1 Weighted Sums 481 25.2 Gauthier's Differential Method 488 26 Fibonometry I 495 26.1 Golden Ratio and Inverse Trigonometric Functions 495 26.2 Golden Triangle Revisited 496 26.3 Golden Weaves 497 26.4 Additional Fibonometric Bridges 498 26.5 Fibonacci and Lucas Factorizations 504 27 Completeness Theorems 509 27.1 Completeness Theorem 509 27.2 Egyptian Algorithm for Multiplication 510 28 The Knapsack Problem 513 28.1 The Knapsack Problem 513 29 Fibonacci and Lucas Subscripts 517 29.1 Fibonacci and Lucas Subscripts 517 29.2 Gibonacci Subscripts 519 29.3 A Recursive Definition of Yn 520 30 Fibonacci and the Complex Plane 525 30.1 Gaussian Numbers 525 30.2 Gaussian Fibonacci and Lucas Numbers 526 30.3 Analytic Extensions 530 1 A.1 Fundamentals 537 SOLUTIONS TO ODD-NUMBERED EXERCISES 575

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詳細情報

• NII書誌ID(NCID): BB25406221
• ISBN:
  • 9781118742129
  • 9781118742082
• LCCN: 2016018243
• 出版国コード: us
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: Hoboken, N.J.
• ページ数/冊数: v.
• 大きさ: 24-25 cm
• 分類:
  • LCC : QA246.5
  • DC23 : 512.7/2
• 件名:
  • LCSH : Fibonacci numbers
  • LCSH : Lucas numbers
• 親書誌ID:
  • BA00007488