Recurrence sequences are of great intrinsic interest and have been a central part of number theory for many years. Moreover, these sequences appear almost everywhere in mathematics and computer science. This book surveys the modern theory of linear recurrence sequences and their generalizations. Particular emphasis is placed on the dramatic impact that sophisticated methods from Diophantine analysis and transcendence theory have had on the subject. Related work on bilinear recurrences and an emerging connection between recurrences and graph theory are covered. Applications and links to other areas of mathematics are described, including combinatorics, dynamical systems and cryptography, and computer science. The book is suitable for researchers interested in number theory, combinatorics, and graph theory.

Readership

Research mathematicians interested in number theory, combinatorics, and graph theory.

Reviews

"The mathematical community should be grateful to the authors for the pains-taking work that they have done, and for the very useful book that they have produced as a result."

-- Bulletin of the London Mathematical Society

"Surprisingly enough, there was no book in the literature entirely devoted to recurrence sequences ... With the book under review, the authors fill this gap in a remarkable way ... this well-written book will be extremely useful for anyone interested in any of the many aspects of linear recurrence sequences."

-- Mathematical Reviews

Table of Contents

• Definitions and techniques
• Zeros, multiplicity and growth
• Periodicity
• Operations on power series and linear recurrence sequences
• Character sums and solutions of congruences
• Arithmetic structure of recurrence sequences
• Distribution in finite fields and residue rings
• Distribution modulo 1 and matrix exponential functions
• Applications to other sequences
• Elliptic divisibility sequences
• Sequences arising in graph theory and dynamics
• Finite fields and algebraic number fields
• Pseudo-random number generators
• Computer science and coding theory
• Appendix: Sequences from the on-line encyclopedia
• Bibliography
• Index