Fibonacci and Lucas numbers with applications / Thomas Koshy, Framingham State University.

Format:

Book

Author/Creator:

Koshy, Thomas, author.

Series:

Pure and applied mathematics (John Wiley & Sons : Unnumbered)

Pure and applied mathematics : a Wiley series of texts, monographs, and tracts

Language:

English

Subjects (All):

Fibonacci numbers.

Lucas numbers.

Physical Description:

volumes : illustrations ; 25 cm.

Edition:

Second edition.

Place of Publication:

Hoboken, New Jersey : John Wiley & Sons, Inc., [2018]-

Summary:

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. Book jacket.

Contents:

31 Fibonacci and Lucas Polynomials I p. 1

31.1 Fibonacci and Lucas Polynomials p. 3

31.2 Pascal's Triangle p. 18

31.3 Additional Explicit Formulas p. 22

31.4 Ends of the Numbers l_n p. 25

31.5 Generating Functions p. 26

31.6 Pell and Pell-Lucas Polynomials p. 27

31.7 Composition of Lucas Polynomials p. 33

31.8 De Moivre-like Formulas p. 35

31.9 Fibonacci-Lucas Bridges p. 36

31.10 Applications of Identity (31.51) p. 37

31.11 Infinite Products p. 48

31.12 Putnam Delight Revisited p. 51

31.13 Infinite Simple Continued Fraction p. 54

32 Fibonacci and Lucas Polynomials II p. 65

32.1 Q-Matrix p. 65

32.2 Summation Formulas p. 67

32.3 Addition Formulas p. 71

32.4 A Recurrence for f₂ p. 76

32.5 Divisibility Properties p. 82

33 Combinatorial Models II p. 87

33.1 A Model for Fibonacci Polynomials p. 87

33.2 Breakability p. 99

33.3 A Ladder Model p. 101

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

33.5 Colored Tilings p. 103

33.6 A New Tiling Scheme p. 104

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

33.8 A Domino Model for Fibonacci Polynomials p. 114

33.9 Another Model for Fibonacci Polynomials p. 118

34 Graph-Theoretic Models II p. 125

34.1 Q-Matrix and Connected Graph p. 125

34.2 Weighted Paths p. 126

34.3 Q-Matrix Revisited p. 127

34.4 Byproducts of the Model p. 128

34.5 A Bijection Algorithm p. 136

34.6 Fibonacci and Lucas Sums p. 137

34.7 Fibonacci Walks p. 140

35 Gibonacci Polynomials p. 145

35.1 Gibonacci Polynomials p. 145

35.2 Differences of Gibonacci Products p. 159

35.3 Generalized Lucas and Ginsburg Identities p. 174

35.4 Gibonacci and Geometry p. 181

35.5 Additional Recurrences p. 184

35.6 Pythagorean Triples p. 188

36 Gibonacci Sums p. 195

36.1 Gibonacci Sums p. 195

36.2 Weighted Sums p. 206

36.3 Exponential Generating Functions p. 209

36.4 Infinite Gibonacci Sums p. 215

37 Additional Gibonacci Delights p. 233

37.1 Some Fundamental Identities Revisited p. 233

37.2 Lucas and Ginsburg Identities Revisited p. 238

37.3 Fibonomial Coefficients p. 247

37.4 Gibonomial Coefficients p. 250

37.5 Additional Identities p. 260

37.6 Strazdins' Identity p. 264

38 Fibonacci and Lucas Polynomials III p. 269

38.1 Seiffert's Formulas p. 270

38.2 Additional Formulas p. 294

38.3 Legendre Polynomials p. 314

39 Gibonacci Determinants p. 321

39.1 A Circulant Determinant p. 321

39.2 A Hybrid Determinant p. 323

39.3 Basin's Determinant p. 333

39.4 Lower Hessenberg Matrices p. 339

39.5 Determinant with a Prescribed First Row p. 343

40 Fibonometry II p. 347

40.1 Fibonometric Results p. 347

40.2 Hyperbolic Functions p. 356

40.3 Inverse Hyperbolic Summation Formulas p. 361

41 Chebyshev Polynomials p. 371

41.1 Chebyshev Polynomials T_n(x) p. 372

41.2 T_n(x) and Trigonometry p. 384

41.3 Hidden Treasures in Table 41.1 p. 386

41.4 Chebyshev Polynomials U_n(x) p. 396

41.5 Pell's Equation p. 398

41.6 U_n(x) and Trigonometry p. 399

41.7 Addition and Cassini-like Formulas p. 401

41.8 Hidden Treasures in Table 41.8 p. 402

41.9 A Chebyshev Bridge p. 404

41.10 T_n and U_n(x) as Products p. 405

41.11 Generating Functions p. 410

42 Chebyshev Tilings p. 415

42.1 Combinatorial Models for U_n p. 415

42.2 Combinatorial Models for T_n p. 420

42.3 Circular Tilings p. 425

43 Bivariate Gibonacci Family I p. 429

43.1 Bivariate Gibonacci Polynomials p. 429

43.2 Bivariate Fibonacci and Lucas Identities p. 430

43.3 Candido's Identity Revisited p. 439

44 Jacobsthal Family p. 443

44.1 Jacobsthal Family p. 444

44.2 Jacobsthal Occurrences p. 450

44.3 Jacobsthal Compositions p. 452

44.4 Triangular Numbers in the Family p. 459

44.5 Formal Languages p. 468

44.6 A USA Olympiad Delight p. 480

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

44.8 Convolutions p. 490

45 Jacobsthal Tilings and Graphs p. 499

45.1 1 × n Tilings p. 499

45.2 2 × n Tilings p. 505

45.3 2 × n Tubular Tilings p. 510

45.4 3 × n Tilings p. 514

45.5 Graph-Theoretic Models p. 518

45.6 Digraph Models p. 522

46 Bivariate Tiling Models p. 537

46.1 A Model for f_n(x, y) p. 537

46.2 Breakability p. 539

46.3 Colored Tilings p. 542

46.4 A Model for l_n(x, y) p. 543

46.5 Colored Tilings Revisited p. 545

46.6 Circular Tilings Again p. 547

47 Vieta Polynomials p. 553

47.1 Vieta Polynomials p. 554

47.2 Aurifeuille's Identity p. 567

47.3 Vieta-Chebyshev Bridges p. 572

47.4 Jacobsthal-Chebyshev Links p. 573

47.5 Two Charming Vieta Identities p. 574

47.6 Tiling Models for V_n p. 576

47.7 Tiling Models for v_n(x) p. 582

48 Bivariate Gibonacci Family II p. 591

48.1 Bivariate Identities p. 591

48.2 Additional Bivariate Identities p. 594

48.3 A Bivariate Lucas Counterpart p. 599

48.4 A Summation Formula for f_2n(x, y) p. 600

48.5 A Summation Formula for l_2n(x, y) p. 602

48.6 Bivariate Fibonacci Links p. 603

48.7 Bivariate Lucas Links p. 606

49 Tribonacci Polynomials p. 611

49.1 Tribonacci Numbers p. 611

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

49.3 Tribonacci Polynomials p. 616

49.4 A Combinatorial Model p. 618

49.5 Tribonacci Polynomials and the Q-Matrix p. 624

49.6 Tribonacci Walks p. 625

49.7 A Bijection Between the Two Models p. 627.

Notes:

Includes bibliographical references and index.

Other Format:

Online version: Koshy, Thomas. Fibonacci and Lucas numbers with applications.

ISBN:

9781118742129

1118742125

9781118742082

1118742087

OCLC:

946788189