Fibonacci and Lucas Numbers with Applications / Edition 1

by Thomas Koshy

Overview

The first comprehensive survey of mathematics' most fascinating number sequences

Fibonacci and Lucas numbers have intrigued amateur and professional mathematicians for centuries. This volume represents the first attempt to compile a definitive history and authoritative analysis of these famous integer sequences, complete with a wealth of exciting applications, enlightening examples, and fun exercises that offer numerous opportunities for exploration and experimentation.

The author has assembled a myriad of fascinating properties of both Fibonacci and Lucas numbers-as developed by a wide range of sources-and catalogued their applications in a multitude of widely varied disciplines such as art, stock market investing, engineering, and neurophysiology. Most of the engaging and delightful material here is easily accessible to college and even high school students, though advanced material is included to challenge more sophisticated Fibonacci enthusiasts. A historical survey of the development of Fibonacci and Lucas numbers, biographical sketches of intriguing personalities involved in developing the subject, and illustrative examples round out this thorough and amusing survey. Most chapters conclude with numeric and theoretical exercises that do not rely on long and tedious proofs of theorems. Highlights include:
* Balanced blend of theory and real-world applications
* Excellent reference material for student reports and projects
* User-friendly, informal, and entertaining writing style
* Historical interjections and short biographies that add a richer perspective to the topic
* Reference sections providing important symbols, problem solutions, and fundamental properties from the theory of numbers and matrices
Fibonacci and Lucas Numbers with Applications provides mathematicians with a wealth of reference material in one convenient volume and presents an in-depth and entertaining resource for enthusiasts at every level and from any background.

Product Details

ISBN-13:	9780471399698
Publisher:	Wiley
Publication date:	10/03/2001
Series:	Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts Series , #51
Pages:	672
Product dimensions:	6.26(w) x 9.45(h) x 1.40(d)

About the Author

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

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 l_n 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 T_n(x) 372

41.2. T_n(x) and Trigonometry 384

41.3. Hidden Treasures in Table 41.1 386

41.4. Chebyshev Polynomials U_n(x) 396

41.5. Pell’s Equation 398

41.6. U_n(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. T_n and U_n as Products 405

41.11. Generating Functions 410

42. Chebyshev Tilings 415

42.1. Combinatorial Models for U_n 415

42.2. Combinatorial Models for T_n 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 𝑓_n(x, y) 537

46.2. Breakability 539

46.3. Colored Tilings 542

46.4. A Model for l_n(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 V_n 576

47.7. Tiling Models for 𝑣_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 𝑓_2n(x, y) 600

48.5. A Summation Formula for l_2n(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

What People are Saying About This

From the Publisher

[Koshy's] book is without doubt the most comprehensive and scholarly work on Fibonacci numbers to date and I am sure that it will quickly signal its presence and impose itself as an authoritative reference manual on Fibonacci numbers.
—Napoleon Gauthier, Royal Military College of Canada, Kingston, ON

What a gem this is! [...] My only regrest about the book is that it wasn't around years ago. It fills such a void.
—Monte Zerger, Adams State College, Alamosa, CO

Editorial Reviews

"...a 'must' for fans of Fibonacci..." (Translated from French, Bulletin AMQ, October 2005)

"...presents a look at the numerous applications of Fibonacci and Lucas numbers in a range of disciplines." (SciTech Book News, Vol. 25, No. 4, December 2001)

"...presents the history, mathematics, and applications of the Fibonacci and Lucas sequences...An important feature of the book is the many...exercises..." (Mathematical Reviews, 2002f)

"...beautiful and well worth the reading...with many exercises and a good bibliography, this book will fascinate both students and teachers." (Mathematics Teacher, Vol. 95, No. 5, May 2002)

"A delightful romp through all things Fibonacci." (American Mathematical Monthly, January 2003)

"...a delightful book which should prove of great value...the most comprehensive collection of results, theorems, and references regarding Fibonacci numbers and their applications to date..." (The Fibonacci Quarterly, February 2002)

"...a splendid compendium of everything that most of us will ever need to know about the Fibonacci and Lucas numbers...an invaluable reference for experts and non-experts alike..." (The Mathematical Gazette)

"...a definitive history and authoritative analysis ...a myriad of fascinating properties of both Fibonacci and Lucas numbers..." (Mathematical Didactics, 2003)

From the Publisher