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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley Series in Pure and Applied Mathematics
- Sprache: Englisch
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.
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Seitenzahl: 613
Veröffentlichungsjahr: 2018
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PURE AND APPLIED MATHEMATICS
A Wiley Series of Texts, Monographs, and Tracts
Founded by RICHARD COURANT
Editors Emeriti: MYRON B. ALLEN III, PETER HILTON, HARRY
HOCHSTADT, ERWIN KREYSZIG, PETER LAX, JOHN TOLAND
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
Copyright
This edition first published 2019
© 2019 John Wiley & Sons, Inc.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of Thomas Koshy to be identified as the author of this work has been asserted in accordance with law.
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The publisher and the authors make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties; including without limitation any implied warranties of fitness for a particular purpose. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for every situation. In view of on‐going research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. The fact that an organization or website is referred to in this work as a citation and/or potential source of further information does not mean that the author or the publisher endorses the information the organization or website may provide or recommendations it may make. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. No warranty may be created or extended by any promotional statements for this work. Neither the publisher nor the author shall be liable for any damages arising herefrom.
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
Symbol
Meaning
or
marginal symbol for alerting the change in notation
unsolved problem
_
end of a proof or solution; end of a lemma, theorem,
or corollary when it does not end in a proof
set of complex numbers
greatest common divisor (gcd) of the positive
integers
least common multiple (lcm) of the positive
integers
mod
remainder when
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.
If I have been able to see farther, it was only
because I stood on the shoulders of giants.
–Sir Isaac Newton (1643–1727)
