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E-grāmata: Fibonacci and Lucas Numbers with Applications, Volume 2 Volume II [Wiley Online]

(Framingham State College)
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Fibonacci and Lucas Numbers with Applications, Volume 2 Volume IIPalielināt
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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.

List of Symbols
xiii
Preface xv
31 Fibonacci and Lucas Polynomials I
1(64)
31.1 Fibonacci and Lucas Polynomials
3(15)
31.2 Pascal's Triangle
18(4)
31.3 Additional Explicit Formulas
22(3)
31.4 Ends of the Numbers ln
25(1)
31.5 Generating Functions
26(1)
31.6 Pell and Pell-Lucas Polynomials
27(6)
31.7 Composition of Lucas Polynomials
33(2)
31.8 De Moivre-like Formulas
35(1)
31.9 Fibonacci--Lucas Bridges
36(1)
31.10 Applications of Identity (31.51)
37(11)
31.11 Infinite Products
48(3)
31.12 Putnam Delight Revisited
51(3)
31.13 Infinite Simple Continued Fraction
54(11)
32 Fibonacci and Lucas Polynomials II
65(22)
32.1 Q-Matrix
65(2)
32.2 Summation Formulas
67(4)
32.3 Addition Formulas
71(5)
32.4 A Recurrence for fn2
76(6)
32.5 Divisibility Properties
82(5)
33 Combinatorial Models II
87(38)
33.1 A Model for Fibonacci Polynomials
87(12)
33.2 Breakability
99(2)
33.3 A Ladder Model
101(1)
33.4 A Model for Pell--Lucas Polynomials: Linear Boards
102(1)
33.5 Colored Tilings
103(1)
33.6 A New Tiling Scheme
104(3)
33.7 A Model for Pell--Lucas Polynomials: Circular Boards
107(7)
33.8 A Domino Model for Fibonacci Polynomials
114(4)
33.9 Another Model for Fibonacci Polynomials
118(7)
34 Graph-Theoretic Models II
125(20)
34.1 Q-Matrix and Connected Graph
125(1)
34.2 Weighted Paths
126(1)
34.3 Q-Matrix Revisited
127(1)
34.4 Byproducts of the Model
128(8)
34.5 A Bijection Algorithm
136(1)
34.6 Fibonacci and Lucas Sums
137(3)
34.7 Fibonacci Walks
140(5)
35 Gibonacci Polynomials
145(50)
35.1 Gibonacci Polynomials
145(14)
35.2 Differences of Gibonacci Products
159(15)
35.3 Generalized Lucas and Ginsburg Identities
174(7)
35.4 Gibonacci and Geometry
181(3)
35.5 Additional Recurrences
184(4)
35.6 Pythagorean Triples
188(7)
36 Gibonacci Sums
195(38)
36.1 Gibonacci Sums
195(11)
36.2 Weighted Sums
206(3)
36.3 Exponential Generating Functions
209(6)
36.4 Infinite Gibonacci Sums
215(18)
37 Additional Gibonacci Delights
233(36)
37.1 Some Fundamental Identities Revisited
233(5)
37.2 Lucas and Ginsburg Identities Revisited
238(9)
37.3 Fibonomial Coefficients
247(3)
37.4 Gibonomial Coefficients
250(10)
37.5 Additional Identities
260(4)
37.6 Strazdins' Identity
264(5)
38 Fibonacci and Lucas Polynomials III
269(52)
38.1 Seiffert's Formulas
270(24)
38.2 Additional Formulas
294(20)
38.3 Legendre Polynomials
314(7)
39 Gibonacci Determinants
321(26)
39.1 A Circulant Determinant
321(2)
39.2 A Hybrid Determinant
323(10)
39.3 Basin's Determinant
333(6)
39.4 Lower Hessenberg Matrices
339(4)
39.5 Determinant with a Prescribed First Row
343(4)
40 Fibonometry II
347(24)
40.1 Fibonometric Results
347(9)
40.2 Hyperbolic Functions
356(5)
40.3 Inverse Hyperbolic Summation Formulas
361(10)
41 Chebyshev Polynomials
371(44)
41.1 Chebyshev Polynomials Tn(x)
372(12)
41.2 Tn(x) and Trigonometry
384(2)
41.3 Hidden Treasures in Table 41.1
386(10)
41.4 Chebyshev Polynomials Un(x)
396(2)
41.5 Pell's Equation
398(1)
41.6 Un(x) and Trigonometry
399(2)
41.7 Addition and Cassini-like Formulas
401(1)
41.8 Hidden Treasures in Table 41.8
402(2)
41.9 A Chebyshev Bridge
404(1)
41.10 Tn and Un as Products
405(5)
41.11 Generating Functions
410(5)
42 Chebyshev Tilings
415(14)
42.1 Combinatorial Models for Un
415(5)
42.2 Combinatorial Models for Tn
420(5)
42.3 Circular Tilings
425(4)
43 Bivariate Gibonacci Family I
429(14)
43.1 Bivariate Gibonacci Polynomials
429(1)
43.2 Bivariate Fibonacci and Lucas Identities
430(9)
43.3 Candido's Identity Revisited
439(4)
44 Jacobsthal Family
443(56)
44.1 Jacobsthal Family
444(6)
44.2 Jacobsthal Occurrences
450(2)
44.3 Jacobsthal Compositions
452(7)
44.4 Triangular Numbers in the Family
459(9)
44.5 Formal Languages
468(12)
44.6 A USA Olympiad Delight
480(3)
44.7 A Story of 1, 2, 7, 42, 429, ...
483(7)
44.8 Convolutions
490(9)
45 Jacobsthal Tilings and Graphs
499(38)
45.1 1 × n Tilings
499(6)
45.2 2 × n Tilings
505(5)
45.3 2 × n Tubular Tilings
510(4)
45.4 3 × n Tilings
514(4)
45.5 Graph-Theoretic Models
518(4)
45.6 Digraph Models
522(15)
46 Bivariate Tiling Models
537(16)
46.1 A Model for fn(x, y)
537(2)
46.2 Breakability
539(3)
46.3 Colored Tilings
542(1)
46.4 A Model for ln(x, y)
543(2)
46.5 Colored Tilings Revisited
545(2)
46.6 Circular Tilings Again
547(6)
47 Vieta Polynomials
553(38)
47.1 Vieta Polynomials
554(13)
47.2 Aurifeuille's Identity
567(5)
47.3 Vieta--Chebyshev Bridges
572(1)
47.4 Jacobsthal-Chebyshev Links
573(1)
47.5 Two Charming Vieta Identities
574(2)
47.6 Tiling Models for Vn
576(6)
47.7 Tiling Models for vn(x)
582(9)
48 Bivariate Gibonacci Family II
591(20)
48.1 Bivariate Identities
591(3)
48.2 Additional Bivariate Identities
594(5)
48.3 A Bivariate Lucas Counterpart
599(1)
48.4 A Summation Formula for f2n(x, y)
600(2)
48.5 A Summation Formula for l2n(x, y)
602(1)
48.6 Bivariate Fibonacci Links
603(3)
48.7 Bivariate Lucas Links
606(5)
49 Tribonacci Polynomials
611(20)
49.1 Tribonacci Numbers
611(2)
49.2 Compositions with Summands 1, 2, and 3
613(3)
49.3 Tribonacci Polynomials
616(2)
49.4 A Combinatorial Model
618(6)
49.5 Tribonacci Polynomials and the g-Matrix
624(1)
49.6 Tribonacci Walks
625(2)
49.7 A Bijection Between the Two Models
627(4)
Appendix 631(13)
A.1 The First 100 Fibonacci and Lucas Numbers
631(3)
A.2 The First 100 Pell and Pell-Lucas Numbers
634(4)
A.3 The First 100 Jacobsthal and Jacobsthal-Lucas Numbers
638(4)
A.4 The First 100 Tribonacci Numbers
642(2)
Abbreviations 644(1)
Bibliography 645(16)
Solutions to Odd-Numbered Exercises 661(64)
Index 725
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
Permanent link: https://www.kriso.lv/db/9781118742297_pe.html
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