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Fibonacci and Lucas numbers with applications /

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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 boo...

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Bibliographic Details
Main Author: Koshy, Thomas (Author)
Format: eBook
Language:English
Published: Hoboken, New Jersey : John Wiley & Sons, Inc., [2018]-
Edition:Second edition.
Series:Pure and applied mathematics (John Wiley & Sons : Unnumbered)
Subjects:
Fibonacci numbers.
Lucas numbers.
Online Access:Connect to the full text of this electronic book
Table of 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<sub>n</sub> 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<sub>n</sub>(x) p. 372
  • 41.2 T<sub>n</sub>(x) and Trigonometry p. 384
  • 41.3 Hidden Treasures in Table 41.1 p. 386
  • 41.4 Chebyshev Polynomials U<sub>n</sub>(x) p. 396
  • 41.5 Pell's Equation p. 398
  • 41.6 U<sub>n</sub>(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<sub>n</sub> and U<sub>n</sub>(x) as Products p. 405
  • 41.11 Generating Functions p. 410
  • 42 Chebyshev Tilings p. 415
  • 42.1 Combinatorial Models for U<sub>n</sub> p. 415
  • 42.2 Combinatorial Models for T<sub>n</sub> 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<sub>n</sub>(x, y) p. 537
  • 46.2 Breakability p. 539
  • 46.3 Colored Tilings p. 542
  • 46.4 A Model for l<sub>n</sub>(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<sub>n</sub> p. 576
  • 47.7 Tiling Models for v<sub>n</sub>(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<sub>2n</sub>(x, y) p. 600
  • 48.5 A Summation Formula for l<sub>2n</sub>(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.