Fibonacci and Lucas Numbers with Applications

By Thomas Koshy

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.

• Language: English
• Publisher: Wiley
• Release date: Dec 14, 2018
• ISBN: 9781118742181

Book preview

Table of Contents

Cover

List of Symbols

Preface

The Twin Shining Stars Revisited

Extended Gibonacci Family

Audience

Prerequisites

Organization

Salient Features

Historical Perspective

Exercises and Solutions

Abbreviations and Symbols Indexes

Appendix

Acknowledgments

31 Fibonacci and Lucas Polynomials I

31.1 Fibonacci and Lucas Polynomials

31.2 Pascal's Triangle

31.3 Additional Explicit Formulas

31.4 Ends of the Numbers

31.5 Generating Functions

31.6 Pell and Pell–Lucas Polynomials

31.7 Composition of Lucas Polynomials

31.8 De Moivre‐Like Formulas

31.9 Fibonacci–Lucas Bridges

31.10 Applications of Identity (31.51)

31.11 Infinite Products

31.12 Putnam Delight Revisited

31.13 Infinite Simple Continued Fraction

32 Fibonacci and Lucas Polynomials II

32.1 Q‐Matrix

32.2 Summation Formulas

32.3 Addition Formulas

32.4 A Recurrence for

32.5 Divisibility Properties

33 Combinatorial Models II

33.1 A Model for Fibonacci Polynomials

33.2 Breakability

33.3 A Ladder Model

33.4 A Model for Pell–Lucas Polynomials: Linear Boards

33.5 Colored Tilings

33.6 A New Tiling Scheme

33.7 A Model for Pell–Lucas Polynomials: Circular Boards

33.8 A Domino Model for Fibonacci Polynomials

33.9 Another Model for Fibonacci Polynomials

34 Graph‐Theoretic Models II

34.1 ‐Matrix and Connected Graph

34.2 Weighted Paths

34.3 ‐Matrix Revisited

34.4 Byproducts of the Model

34.5 A Bijection Algorithm

34.6 Fibonacci and Lucas Sums

34.7 Fibonacci Walks

35 Gibonacci Polynomials

35.1 Gibonacci Polynomials

35.2 Differences of Gibonacci Products

35.3 Generalized Lucas and Ginsburg Identities

35.4 Gibonacci and Geometry

35.5 Additional Recurrences

35.6 Pythagorean Triples

36 Gibonacci Sums

36.1 Gibonacci Sums

36.2 Weighted Sums

36.3 Exponential Generating Functions

36.4 Infinite Gibonacci Sums

37 Additional Gibonacci Delights

37.1 Some Fundamental Identities Revisited

37.2 Lucas and Ginsburg Identities Revisited

37.3 Fibonomial Coefficients

37.4 Gibonomial Coefficients

37.5 Additional Identities

37.6 Strazdins' Identity

38 Fibonacci and Lucas Polynomials III

38.1 Seiffert's Formulas

38.2 Additional Formulas

38.3 Legendre Polynomials

39 Gibonacci Determinants

39.1 A Circulant Determinant

39.2 A Hybrid Determinant

39.3 Basin's Determinant

39.4 Lower Hessenberg Matrices

39.5 Determinant with a Prescribed First Row

40 Fibonometry II

40.1 Fibonometric Results

40.2 Hyperbolic Functions

40.3 Inverse Hyperbolic Summation Formulas

41 Chebyshev Polynomials

41.1 Chebyshev Polynomials

41.2 and Trigonometry

41.3 Hidden Treasures in Table 41.1

41.4 Chebyshev Polynomials

41.5 Pell's Equation

41.6 and Trigonometry

41.7 Addition and Cassini‐like Formulas

41.8 Hidden Treasures in Table 41.8

41.9 A Chebyshev Bridge

41.10 and as Products

41.11 Generating Functions

Chapter 42: Chebyshev Tilings

42.1 Combinatorial Models for

42.2 Combinatorial Models for

42.3 Circular Tilings

43 Bivariate Gibonacci Family I

43.1 Bivariate Gibonacci Polynomials

43.2 Bivariate Fibonacci and Lucas Identities

43.3 Candido's Identity Revisited

44 Jacobsthal Family

44.1 Jacobsthal Family

44.2 Jacobsthal Occurrences

44.3 Jacobsthal Compositions

44.4 Triangular Numbers in the Family

44.5 Formal Languages

44.6 A USA Olympiad Delight

44.7 A Story of 1, 2, 7, 42, 429,

44.8 Convolutions

45 Jacobsthal Tilings and Graphs

45.1 Tilings

45.2 Tilings

45.3 Tubular Tilings

45.4 Tilings

45.5 Graph‐Theoretic Models

45.6 Digraph Models

Exercises 45

46 Bivariate Tiling Models

46.1 A Model for

46.2 Breakability

46.3 Colored Tilings

46.4 A Model for

46.5 Colored Tilings Revisited

46.6 Circular Tilings Again

46.7 Exercises 46

47 Vieta Polynomials

47.1 Vieta Polynomials

47.2 Aurifeuille s Identity

47.3 Vieta–Chebyshev Bridges

47.4 Jacobsthal–Chebyshev Links

47.5 Two Charming Vieta Identities

47.6 Tiling Models for

47.7 Tiling Models for

47.8 Exercises 47

48 Bivariate Gibonacci Family II

48.1 Bivariate Identities

48.2 Additional Bivariate Identities

48.3 A Bivariate Lucas Counterpart

48.4 A Summation Formula for

48.5 A Summation Formula for

48.6 Bivariate Fibonacci Links

48.7 Bivariate Lucas Links

48.8 Exercises 48

49 Tribonacci Polynomials

49.1 Tribonacci Numbers

49.2 Compositions with Summands 1, 2, and 3

49.3 Tribonacci Polynomials

49.4 A Combinatorial Model

49.5 Tribonacci Polynomials and the ‐Matrix

49.6 Tribonacci Walks

49.7 A Bijection Between the Two Models

Appendix

Abbreviations

Bibliography

Solutions to Odd‐Numbered Exercises

Exercises 31

Exercises 32

Exercises 33

Exercises 34

Exercises 35

Exercises 36

Exercises 37

Exercises 38

Exercises 39

Exercises 40

Exercises 41

Exercises 42

Exercises 43

Exercises 44

Exercises 45

Exercises 46

Exercises 47

Exercises 48

Exercises 49

Index

End User License Agreement

List of Tables

Chapter 31

Table 31.1 First 10 Fibonacci and Lucas Polynomials

Table 31.2 Triangular Array

Table 31.4

Table 31.5

Table 31.6 First 10 Pell and Pell–Lucas Polynomials

Table 31.7 First 10 Pell and Pell–Lucas Numbers

Chapter 33

Table 33.1

Table 33.2

Table 33.3

Table 33.4 Set and , Where

Table 33.5 Sets and Their Weights, Where

Table 33.6

Table 33.7

Table 33.8

Chapter 34

Table 34.1 Paths of Length 4

Table 34.2 Closed Paths of Length 5 from to

Table 34.3

Table 34.4

Table 34.5

Table 34.6

Table 34.7

Table 34.8 Closed Paths from to and the Corresponding Fibonacci Tilings

Table 34.9 Paths of Length from to

Table 34.10 Paths of Even Length from  to 

Table 34.11 Paths of Even Length from  to 

Chapter 36

Table 36.1 Array

Table 36.2 Array

Table 36.3

Table 36.4 Array

Table 36.5 Array

Table 36.6

Table 36.7 Array

Chapter 37

Table 37.1 Fibonomial Triangle

Table 37.2 Gibonomial Triangle

Table 37.3

Chapter 38

Table 38.1

Table 38.2

Table 38.3 First Eight Legendre Polynomials

Chapter 41

Table 41.1 Chebyshev Polynomials

Table 41.2

Table 41.3 Array

Table 41.4

Table 41.5

Table 41.6

Table 41.7 Triangular Array

Table 41.8

Table 41.9

Table 41.10 Triangular Array

Chapter 42

Table 42.1

Chapter 43

Table 43.1 First 10 Bivariate Fibonacci and Lucas Polynomials

Chapter 44

Table 44.1 First 10 Jacobsthal and Jacobsthal–Lucas Polynomials

Table 44.2 First 10 Jacobsthal and Jacobsthal–Lucas Numbers

Table 44.3

Table 44.4 Compositions with 1s, 2s, and s

Table 44.5 Compositions with Last Summand Odd

Table 44.6 Values of , where

Table 44.7 Compositions with Last Summand Even

Table 44.8 First 15 Triangular Numbers

Table 44.9

Table 44.10

Table 44.11

Table 44.12

Table 44.13

Table 44.14

Table 44.15

Table 44.16 Words of Length

Table 44.17 Values of , and

Table 44.18 Ternary Words and Their Counts

Table 44.19 Possible Paths

Table 44.20 The Seven ASMs

Table 44.21 Values of , where

Table 44.22 Values of , where

Chapter 45

Table 45.1

Table 45.2 Values of , where

Table 45.3 Jacobsthal Polynomials

Table 45.4 Jacobsthal Numbers

Table 45.5 Closed Paths of Length 4 at

Table 45.6 Closed Paths of Length 4 at and

Table 45.7 Directed Paths of Length 4

Chapter 47

Table 47.1 First 10 Vieta and Vieta–Lucas Polynomials

Chapter 49

Table 49.1

Table 49.2

Table 49.3 First 10 Tribonacci Polynomials

Table 49.4 Tribonacci Array

Table 49.5

Table 49.6 Tribonacci Weights of Length 9

Table 49.7

Table 49.8

Table 49.9

Table 49.10 Closed Walks of Length 4 and the Corresponding Tribonacci Tilings

List of Illustrations

Chapter 31

Figure 31.1

Figure 31.2

Figure 31.3

Figure 31.4Figure 31.4 Pascal's Triangle.

Figure 31.5

Figure 31.6Figure 31.6 Pascal's Triangle.

Figure 31.7

Figure 31.8

Chapter 33

Figure 33.1 Tilings of a board, where .

Figure 33.2

Figure 33.3

Figure 33.4

Figure 33.5

Figure 33.6

Figure 33.7

Figure 33.8

Figure 33.9

Figure 33.10

Figure 33.11

Figure 33.12

Figure 33.13

Figure 33.14

Figure 33.15

Figure 33.16

Figure 33.17

Figure 33.18

Figure 33.19

Figure 33.20

Figure 33.21

Figure 33.22

Figure 33.23

Figure 33.24

Figure 33.25

Figure 33.26

Figure 33.27 ‐bracelets, where .

Figure 33.28

Figure 33.29 5‐bracelets.

Figure 33.30 3‐bracelets.

Figure 33.31

Figure 33.32

Figure 33.33

Figure 33.34

Figure 33.35

Figure 33.36

Figure 33.37

Figure 33.38 Tilings of a board and the corresponding pairs .

Figure 33.39

Figure 33.40

Figure 33.42

Figure 33.43

Figure 33.44

Figure 33.45

Figure 33.46

Figure 33.47 Domino tilings of a board.

Figure 33.48 Domino tilings of a board.

Figure 33.49 A tiling.

Figure 33.50 tilings breakable at cell 4.

Figure 33.51 tilings unbreakable at cell 4.

Figure 33.52 tilings.

Figure 33.53 Sequences of differences of elements of and tilings in .

Figure 33.54 tilings and the corresponding elements of .

Chapter 34

Figure 34.1 Weighted digraph.

Figure 34.2 Fibonacci walks.

Figure 34.3 Fibonacci walks of length 3.

Chapter 35

Figure 35.2 .

Figure 35.3 .

Figure 35.4 .

Chapter 37

Figure 37.1 A binary version of the fibonomial triangle.

Figure 37.2 Star of David.

Figure 37.3 Products of triples equal.

Figure 37.4 Products of triples = 27,442,800.

Chapter 41

Figure 41.1

Figure 41.2

Figure 41.3

Figure 41.4 Triangular array .

Chapter 42

Figure 42.1 Chebyshev models for using Model I.

Figure 42.2 Chebyshev tilings of length 5.

Figure 42.3 Chebyshev models for using Model II.

Figure 42.4 Chebyshev models for using Model III.

Figure 42.5 Colored tilings of length using Model IV.

Figure 42.6 Tilings of length 4 using Model II.

Figure 42.7 Tilings of length 4 beginning with a white square.

Figure 42.8 Tilings of length 4 not beginning with a white square.

Figure 42.9

Chapter 43

Figure 43.1 Pascal's triangle.

Chapter 44

Figure 44.1

Figure 44.2

Figure 44.3

Figure 44.4

Figure 44.5 Number of ASMs: triangular array .

Chapter 45

Figure 45.1

Figure 45.2 Weighted Pascal's triangle.

Figure 45.3

Figure 45.4 Tilings of a board, where .

Figure 45.5 Southeast diagonal sums.

Figure 45.6 Northeast diagonal sums.

Figure 45.7 Tilings of a board, where .

Figure 45.8

Figure 45.9 A bijective proof without words.

Figure 45.10Figure 45.10 A complete graph .

Figure 45.11 Path graph with a loop at .

Figure 45.12 Weighted digraph.

Figure 45.13 A digraph with four edges.

Figure 45.14 A digraph model for the Jacobsthal family.

Figure 45.15

Chapter 46

Figure 46.1

Figure 46.2

Figure 46.3

Figure 46.4

Figure 46.5

Figure 46.6

Figure 46.7

Chapter 47

Figure 47.1 A modified Pascal s triangle.

Figure 47.2 Tiling models for using Model I.

Figure 47.3 Tilings of length 3.

Figure 47.4 Tilings of length 4.

Figure 47.5 Tiling models for using Model II.

Figure 47.6 Domino tilings of a board.

Figure 47.7 Domino tilings of a board.

Figure 47.8 Tiling models for using Model IV.

Figure 47.9 Tilings of length 4.

Figure 47.10 Circular tilings.

Chapter 48

Figure 48.1

Figure 48.2

Chapter 49

Figure 49.1 Tribonacci array.

Figure 49.2 A complete 3‐ary rooted tree.

Figure 49.3 Tribonacci tilings of length 4.

Figure 49.4 Tilings with an odd number of squares.

Figure 49.5 Tilings with an odd number of triminoes.

Figure 49.6 A tribonacci digraph.

Figure 49.7 Array .

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A complete list of the titles in this series appears at the end of this volume.

Fibonacci and Lucas Numbers with Applications

Volume Two

Thomas Koshy

Framingham State University

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Library of Congress Cataloging‐in‐Publication Data

Names: Koshy, Thomas.

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

Description: Second edition. | Hoboken, New Jersey : John Wiley & Sons, Inc., [2019]‐ | Series: Pure and applied mathematics: a Wiley series of texts, monographs, and tracts | Includes bibliographical references and index.

Identifiers: LCCN 2016018243 | ISBN 9781118742082 (cloth : v. 2)

Subjects: LCSH: Fibonacci numbers. | Lucas numbers.

Classification: LCC QA246.5 .K67 2019 | DDC 512.7/2–dc23 LC record available at https://lccn.loc.gov/2016018243

Cover image: © NDogan/Shutterstock

Cover design by Wiley

Dedication

Dedicated to

the loving memory of

Dr. Kolathu Mathew Alexander

(1930-2017)

List of Symbols

Preface

Man has the faculty of becoming completely absorbed in one subject, no matter how trivial, and no subject is so trivial that it will not assume infinite proportions if one's entire attention is devoted to it.

–Tolstoy, War and Peace

The Twin Shining Stars Revisited

The main focus of Volume One was to showcase the beauty, applications, and ubiquity of Fibonacci and Lucas numbers in many areas of human endeavor. Although these numbers have been investigated for centuries, they continue to charm both creative amateurs and mathematicians alike, and provide exciting new tools for expanding the frontiers of mathematical study. In addition to being great fun, they also stimulate our curiosity and sharpen mathematical skills such as pattern recognition, conjecturing, proof techniques, and problem‐solving. The area is still so fertile that growth opportunities appear to be endless.

Extended Gibonacci Family

The gibonacci numbers in Chapter 7 provide a unified approach to Fibonacci and Lucas numbers. In a similar way, we can extend these twin numeric families to twin polynomial families. For the first time, the present volume extends the gibonacci polynomial family even further. Besides Fibonacci and Lucas polynomials and their numeric counterparts, the extended gibonacci family includes Pell, Pell–Lucas, Jacobsthal, Jacobsthal–Lucas, Chebyshev, and Vieta polynomials, and their numeric counterparts as subfamilies. This unified approach gives a comprehensive view of a very large family of polynomial functions, and the fascinating relationships among the subfamilies. The present volume provides the largest and most extensive study of this spectacular area of discrete mathematics to date.

Over the years, I have had the privilege of hearing from many Fibonacci enthusiasts around the world. Their interest gave me the strength and courage to embark on this massive task.

Audience

The present volume, which is a continuation of Volume One, is intended for a wide audience, including professional mathematicians, physicists, engineers, and creative amateurs. It provides numerous delightful opportunities for proposing and solving problems, as well as material for talks, seminars, group discussions, essays, applications, and extending known facts.

This volume is the result of extensive research using over 520 references, which are listed in the bibliography. It should serve as an invaluable resource for Fibonacci enthusiasts in many fields. It is my sincere hope that this volume will aid them in exploring this exciting field, and in advancing the boundaries of our current knowledge with great enthusiasm and satisfaction.

Prerequisites

A familiarity with the fundamental properties of Fibonacci and Lucas numbers, as in Volume One, is an indispensable prerequisite. So is a basic knowledge of combinatorics, generating functions, graph theory, linear algebra, number theory, recursion, techniques of solving recurrences, and trigonometry.

Organization

The book is divided into 19 chapters of manageable size. Chapters 31 and 32 present an extensive study of Fibonacci and Lucas polynomials, including a continuing discussion of Pell and Pell–Lucas polynomials. They are followed by combinatorial and graph‐theoretic models for them in Chapters 33 and 34. Chapters 35–39 offer additional properties of gibonacci polynomials, followed in Chapter 40 by a blend of trigonometry and gibonacci polynomials. Chapters 41 and 42 deal with a short introduction to Chebyshev polynomials and combinatorial models for them. Chapters 44 and 45 are two delightful studies of Jacobsthal and Jacobsthal–Lucas polynomials, and their numeric counterparts. Chapters 43, 46, and 48 contain a short discussion of bivariate gibonacci polynomials and their combinatorial models. Chapter 47 gives a brief discourse on Vieta polynomials, combinatorial models, and the relationships among the gibonacci subfamilies. Chapter 49 presents tribonacci numbers and polynomials; it also highlights their combinatorial and graph‐theoretic models.

Salient Features

This volume, like Volume One, emphasizes a user‐friendly and historical approach; it includes a wealth of applications, examples, and exercises; numerous identities of varying degrees of sophistication; current applications and examples; combinatorial and graph‐theoretic models; geometric interpretations; and links among and applications of gibonacci subfamilies.

Historical Perspective

As in Volume One, I have made every attempt to present the material in a historical context, including the name and affiliation of every contributor, and the year of the contribution; indirectly, this puts a human face behind each discovery. I have also included photographs of some mathematicians who have made significant contributions to this ever‐growing field.

Again, my apologies to those contributors whose names or affiliations are missing; I would be grateful to hear about any omissions.

Exercises and Solutions

The book features over 1,230 exercises of varying degrees of difficulty. I encourage students and Fibonacci enthusiasts to have fun with them; they may open new avenues for further exploration. Abbreviated solutions to all odd‐numbered exercises are given at the end of the book.

Abbreviations and Symbols Indexes

An updated list of symbols, standard and nonstandard, appears in the front of the book. In addition, I have used a number of abbreviations in the interest of brevity; they are listed at the end of the book.

Appendix

The Appendix contains four tables: the first 100 Fibonacci and Lucas numbers; the first 100 Pell and Pell–Lucas numbers; the first 100 Jacobsthal and Jacobsthal–Lucas numbers; and a table of 100 tribonacci numbers. These should be useful for hand computations.

Acknowledgments

A massive project such as this is not possible without constructive input from a number of sources. I am grateful to all those who played a significant role in enhancing the quality of the manuscript with their thoughts, suggestions, and comments.

My gratitude also goes to George E. Andrews, Marjorie Bicknell‐Johnson, Ralph P. Grimaldi, R.S. Melham, and M.N.S. Swamy for sharing their brief biographies and photographs; to Margarite Landry for her superb editorial assistance; to Zhenguang Gao for preparing the tables in the Appendix; and to the staff at John Wiley & Sons, especially Susanne Steitz (former mathematics editor), Kathleen Pagliaro, and Jon Gurstelle for their enthusiasm and confidence in this huge endeavor.

Finally, I would be grateful to hear from readers about any inadvertent errors or typos, and especially delighted to hear from anyone who has discovered new properties or applications.

Thomas Koshy

[e-mail]

Framingham, Massachusetts

August, 2018

If I have been able to see farther, it was only

because I stood on the shoulders of giants.

–Sir Isaac Newton (1643–1727)

31

Fibonacci and Lucas Polynomials I

A man may die,

nations may rise and fall,

but an idea lives on.

–John F. Kennedy (1917–1963)

The celebrated Fibonacci polynomials were originally studied beginning in 1883 by the Belgian mathematician Eugene C. Catalan, and later by the German mathematician Ernst Jacobsthal (1882–1965). They were further investigated by M.N.S. Swamy at the University of Saskatchewan, Canada. The equally famous Lucas polynomials were studied beginning in 1970 by Marjorie Bicknell of Santa Clara, California [37].

Photo of Eugène Charles Catalan (1814-1894).

Eugène Charles Catalan (1814–1894) was born in Bruges, Belgium, and received his Doctor of Science from the École Polytechnique in Paris. After working briefly at the Department of Bridges and Highways, he became professor of mathematics at Collège de Chalons‐sur‐Marne, and then at Collège Charlemagne. Catalan went on to teach at Lycée Saint Louis. In 1865, he became professor of analysis at the University of Liège. He published Éléments de Géométrie (1843) and Notions d'astronomie (1860), as well as many articles on multiple integrals, the theory of surfaces, mathematical analysis, calculus of probability, and geometry. Catalan is well known for extensive research on spherical harmonics, analysis of differential equations, transformation of variables in multiple integrals, continued fractions, series, and infinite products.

Portrait of M.N.S. Swamy.

M.N.S. Swamy was born in Karnataka, India. He received his B.Sc. (Hons) in Mathematics from Mysore University in 1954; Diploma in Electrical Engineering from the Indian Institute of Science, Bangalore, in 1957; and M.Sc. (1960) and Ph.D. (1963) in Electrical Engineering from the University of Saskatchewan, Canada.

A former Chair of the Department of Electrical Engineering and Dean of Engineering and Computer Science at Concordia University, Canada, Swamy is currently a Research Professor and the Director of the Center for Signal Processing and Communications. He has also taught at the Technical University of Nova Scotia, and the Universities of Calgary and Saskatchewan.

Swamy is a prolific problem‐proposer and problem‐solver well known to the Fibonacci audience. He has published extensively in number theory, circuits, systems, and signal processing and has written three books. He is the editor‐in‐chief of Circuits, Systems, and Signal Processing, and an associate editor of The Fibonacci Quarterly, and a sustaining member of the Fibonacci Association.

Swamy received the Commemorative Medal for the 125th Anniversary of the Confederation of Canada in 1993 in recognition of his significant contributions to Canada. In 2001, he was awarded D.Sc. in Engineering by Ansted University, British Virgin Islands, in recognition of his exemplary contributions to the research in Electrical and Computer Engineering and to Engineering Education, as well as his dedication to the promotion of Signal Processing and Communications Applications.

Portrait of Marjorie Bicknell-Johnson.

Marjorie Bicknell‐Johnson was born in Santa Rosa, California. She received her B.S. (1962) and M.A. (1964) in Mathematics from San Jose State University, California, where she wrote her Master's thesis, The Lambda Number of a Matrix, under the guidance of V.E. Hoggatt, Jr.

The concept of the lambda number of a matrix first appears in the unpublished notes of Fenton S. Stancliff (1895–1962) of Meadville, Pennsylvania. (He died in Springfield, Ohio in 1962.) His extensive notes are pages of numerical examples without proofs or coherent definitions, that provided material for further study. Bicknell developed the mathematics of the lambda function in her thesis [40].

A charter member of the Fibonacci Association, Bicknell‐Johnson has been a member of its Board of Directors since 1967, as well as Secretary (1965–2010) and Treasurer (1981–1999). In 2012, she wrote a history of the first 50 years of the Association [39].

Bicknell‐Johnson has been a passionate and enthusiastic contributor to the world of Fibonacci and Lucas numbers, as author or co‐author of research papers, 32 of them written with Hoggatt. Her 1980 obituary of Hoggatt remains a fine testimonial to their productive association [38].

31.1 Fibonacci and Lucas Polynomials

As we might expect, they satisfy the same polynomial recurrence , where . When and ; and when and . Table 31.1 gives the first ten Fibonacci and Lucas polynomials in . Clearly, and .

Table 31.1 First 10 Fibonacci and Lucas Polynomials

In the interest of brevity and clarity, we drop the argument in the functional notation, when such deletions do notcause any confusion. Thus will mean , although is technically a functional name and notan output value.

For the curious‐minded, we add that is an even function when is odd, and an odd function when is even; and is an odd function when is odd, and even when is even.

Table 31.2 Triangular Array

ntb001

Table 31.1 contains some hidden treasures. To see them, we arrange the nonzero coefficients of the Fibonacci polynomials in a left‐justified array ; see Table 31.2. Column 2 of the array consists of the triangular numbers , and the th row sum is .

Let denote the element in row and column of the array. Clearly, is the coefficient of in ; so . Recall that [287].

Consequently, it can be defined recursively:

equation

where and ; see the arrows in Table 31.2. This can be confirmed; see Exercise 31.1.

Let denote the th rising diagonal sum. The sequence shows an interesting pattern: 1, 1, 1, 2, 3, 4, ⑥, 9, 13, …; see Figure 31.1 We can also define recursively:

equation

where .

Image described by surrounding text.

Figure 31.1

Since , it follows that

equation

For example, .

The falling diagonal sums also exhibit an interesting pattern: 1, 2, 4, 8, 16, …; see Figure 31.2. This is so, since the th such sum is given by

equation

where .

Image described by surrounding text.

Figure 31.2

The nonzero elements of Lucas polynomials also manifest interesting properties; see array in Table 31.3.

Table 31.3 Triangular Array

ntb002

Let denote the element in row and column , where and . Then

1) .

2) , where , and .

3) Let denote the th rising diagonal sum. Then , and , where .

4) .

For example,

.

In the interest of brevity, we omit their proofs; see Exercises 31.2–31.5.

Next we construct a graph‐theoretic model for Fibonacci polynomials.

Weighted Fibonacci Trees

Recall from Chapter 4 that the th Fibonacci tree is a (rooted) binary tree [287]such that

1) both and consist of exactly one vertex; and

2) is a binary tree whose left subtree is and right subtree is , where . It has vertices, leaves, internal vertices, and edges.

Figure 31.3 shows the first five Fibonacci trees.

First five Fibonacci trees labeled T1, T2, T3, T4, and T5 (left-right).

Figure 31.3

We now assign a weightto recursively. The weight of is 1 and that of is . Then the weight of is defined by , where .

For example, ; and .

Since , and , it follows by the recursive definition that , where . Clearly, gives the number of leaves of when .

Binet‐like Formulas

Using the recurrence and the initial conditions, we can derive explicit formulas for both and ; see Exercises 31.6 and 31.7:

equation

where and are the solutions of the equation and . Notice that , and .

Since and , it follows by the principle of mathematical induction (PMI) that , where ; see Exercise 31.8. Similarly .

Using the Binet‐like formulas, we can extend the definitions of Fibonacci and Lucas polynomials to negative subscripts: and .

Using the Binet‐like formulas, we can also extract a plethora of properties of Fibonacci and Lucas polynomials; see Exercises 31.14–31.97. For example, it is fairly easy to establish that

31.1 equation

(31.2) equation

The last two identities are Cassini‐like formulas. It follows from the Cassini‐like formula for that every two consecutive Fibonacci polynomials are relatively prime; that is, , where denotes the greatest common divisor (gcd) of the polynomials and .

Cassini‐like Formulas Revisited

Since , it follows that , where . Consequently, by the Cassini‐like formula for , every two consecutive Lucas polynomials are relatively prime, that is, .

The Cassini‐like formulas have added dividends. For instance, . To see this, we have

equation

Replacing with , this implies . Thus

31.3

equation

Similarly,

(31.4) equation

see Exercise 31.102.

It follows from properties 31.3and 31.4that

equation

For example,

.

Pythagorean Triples

The identities and (see Exercises 31.32 and 31.49) can be employed to construct Pythagorean triples . To see this, let and . We now find such that is a Pythagorean triple.

Since , we have

equation

Therefore,

; so we obtain .

Thus

is a Pythagorean triple.

Clearly, and ; so . Consequently, and hence . Thus is nota primitive Pythagorean triple.

H.T. Freitag (1908–2005) of Roanoke, Virginia, studied the Pythagorean triple for the special case in 1991 [168].

Recall from Chapter 16 that . So what can we say about ? Next we investigate this.

Suppose . Then . Since . Consequently,

equation

Similarly, . Thus

(31.5) equation

where .

For the curious‐minded, we add that

equation

It follows by the recursive definition that deg( ) = and deg( ) = , where deg( ) denotes the degree of the polynomial and . Suppose . Then ; consequently, . Suppose also that . Since deg( ) = deg( ) + deg( ) = , it follows that . Likewise, .

The facts that , and can be used to develop two interesting identities, one involving Fibonacci polynomials and the other involving Lucas polynomials.

To begin, we have

(31.6) equation

Similarly,

(31.7) equation

It follows by equations 31.6and 31.7that

(31.8) equation

(31.9) equation

see Exercise 31.71.

For example,

equation

Identity 31.8, in particular, yields

equation

J.L. Brown of Pennsylvania State University found this result in 1965 [59].

The next example is an interesting application of identity 31.1.

Example 31.1

Prove that

equation

Proof

Suppose is even. Let . Using identity 31.1, we can rewrite as a telescoping sum:

equation

This yields the desired formulas when is even.

The formulas when is odd follow similarly; see Exercise 31.72. __

In 1996, R. Euler of Northwest Missouri State University studied this example for the case [152].

In particular, let . Then

equation

Generalized Cassini‐like Formulas

The Cassini‐like formulas can be generalized as follows:

(31.10) equation

(31.11)

equation

see Exercises 31.73 and 31.74.

It follows that both and are divisible by . In particular, both and are divisible by . It also follows that and .

It follows from identities 31.10and 31.11that

(31.12) equation

(31.13) equation

Identity 31.12is a generalization of the d'Ocagne identity , named after the French mathematician Philbert Maurice d'Ocagne (1862–1938).

It also follows from identities 31.10and 31.11that

(31.14) equation

(31.15) equation

see Exercises 31.76 and 31.77.

These two identities imply that

(31.16) equation

(31.17) equation

(31.18) equation

see Exercises 31.78–31.80. Consequently, .

The next example features a neat application of identity 31.10. It was originally studied in 1969 by Swamy [489].

Example 31.2

Prove that

equation

Proof

It follows from identity 31.10that

(31.19) equation

Consequently, we have

(31.20) equation

It also follows by identity 31.10that

31.21

equation

The given result now follows by equations 31.20and 31.21. __

The formula in Example 2has a Lucas counterpart:

equation

see Exercise 31.148.

A quick look at Table 31.1 reveals that the constant term in is 1 if is odd, and 0 if is even; and the constant term in is 0 if is odd, and 2 if is even. We now confirm these observations.

Ends of the Polynomials and

Since . Therefore, by the Binet‐like formula for . So ends in 1 if is odd, and 0 if is even.

On the other hand, let

equation

Then . So ends in 0 if is odd, and 2 otherwise.

Next we develop two bridges linking and , by employing a bit of differential and integral calculus.

Links Between and

Since , it follows that and , where the prime denotes differentiation with respect to . By the Binet‐like formula for , we then have

31.22 equation

It follows from identity 31.22that .

For example, ; so ; and .

To see a related link, property 31.22implies that we can recover from by integrating both sides from 0 to :

31.23 equation

It follows from 31.23that

(31.24) equation

For example,

; and

. Clearly,

.

We can use the Binet‐like formula for , coupled