This self-contained text presents state-of-the-art results on recurrent
sequences and their applications in algebra, number theory, geometry of
the complex plane and discrete mathematics. It is designed to appeal to
a wide readership, ranging from scholars and academics, to undergraduate
students, or advanced high school and college students training for
competitions. The content of the book is very recent, and focuses on
areas where significant research is currently taking place. Among the
new approaches promoted in this book, the authors highlight the
visualization of some recurrences in the complex plane, the concurrent
use of algebraic, arithmetic, and trigonometric perspectives on
classical number sequences, and links to many applications. It contains
techniques which are fundamental in other areas of math and encourages
further research on the topic. The introductory chapters only require
good understanding of college algebra, complex numbers, analysis and
basic combinatorics. For Chapters 3, 4 and 6 the prerequisites include
number theory, linear algebra and complex analysis.
The first part of the book presents key theoretical elements required
for a good understanding of the topic. The exposition moves on to to
fundamental results and key examples of recurrences and their
properties. The geometry of linear recurrences in the complex plane is
presented in detail through numerous diagrams, which lead to often
unexpected connections to combinatorics, number theory, integer
sequences, and random number generation. The second part of the book
presents a collection of 123 problems with full solutions, illustrating
the wide range of topics where recurrent sequences can be found. This
material is ideal for consolidating the theoretical knowledge and for
preparing students for Olympiads.
