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Fibonacci and Lucas Numbers with Applications

Thomas Koshy
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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These two volumes give an encyclopedic coverage of Fibonacci and Lucas numbers. Most mathematicians are familiar with the Fibonacci numbers, defined by the recurrence \( F_{n+2}=F_{n+1}+F_{n} \) with \( F_{0}=0 \) and \( F_{1}=1\). The Lucas numbers are less familiar; they are defined by the same recurrence \( L_{n+2}=L_{n+1}+L_{n} \) with the different initial values \( L_{0}=2 \) and \( L_{1}=1 \).  The Golden Ratio \( \sqrt{5} + 1)/2 = 1.61803 \ldots \) is key to understanding these sequences and gets a lot of coverage. Usually the Golden Ratio is denoted by \( \phi \) but these books use \( \alpha \).
The present work is an expansion of the first edition, published in 2001 at half the number of pages. (Bibliographic note: only Volume 1 of this new edition is marked “Second Edition”.)  The books are both comprehensive and discursive. They include vast numbers of facts (mostly formulas) about the Fibonacci and Lucas numbers, and they include discussions of any tangible or intangible things where these numbers or the Golden Ratio appear, even if there is no other mathematical content to them. These are the Applications of the title. The also cover Fibonacci-like sequences such as the Gibonacci (Generalized Fibonacci) numbers, that satisfy the same recurrence with arbitrary initial values.
I view these books primarily as references, but they are structured so that they can be textbooks also. They include on the order of 2,000 exercises, most of them to prove formulas, and there are brief answers to the odd-numbered exercises in the back of the book. Most of the material is easy to understand; it could be used in an upper-division undergraduate or beginning graduate course.
Much of the work in this area tends to be ad hoc, using the defining recurrence in clever ways. One strength of the present books is that they attempt to be more systematic and make use of more powerful standard methods from linear difference equations. These methods include Binet’s formula (based on fundamental solutions), generating functions, and matrix methods. The last method is not as well known as it should be, and happily, it gets thorough coverage here. The main matrix is the \( Q \)-matrix, defined by
\( Q=\left[ \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right] \)
Many Fibonacci and Lucas formulas can be developed from powers of the \( Q \)-matrix multiplied by appropriate vectors.
The second volume in particular makes much use of “polynomial methods”, where we generalize for example the Fibonacci numbers to the Fibonacci polynomials defined by the recurrence \( f_{n}(x)=xf_{n-1}(x)+f_{n-2}(x) \). These polynomials are closely related to the Chebyshev polynomials and different sequences of polynomials count different types of combinatorial objects. Thus they are useful more for counting than for proving Fibonacci identities.
A nice feature is photos and brief biographers of the leading researchers in this field, past and present. There are reasonably-complete indices to each volume.
Other useful books on the Fibonacci numbers include Vajda’s Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications and Vorobiev’s Fibonacci Numbers. Vajda’s book is close in spirit to the present book; it gives broad coverage of the same topics but does not go into depth.  There are also no exercises. Vorobiev’s book is slanted away from identities and toward number-theoretic and other properties of the Fibonacci numbers; thus it has a lot about divisibility and about continued fractions; it also has no exercises. Another book that I have not seen but has favorable reviews is Grimaldi’s Fibonacci and Catalan Numbers: An Introduction. It is intended as a textbook and has about 300 exercises. The scholarly journal Fibonacci Quarterly is a valuable source of current research; most of the results in these books originated there.


Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web site is His mathematical interests are number theory and classical analysis.