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.
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Definitions and Techniques
Zeros Multiplicity and Growth
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
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algebraic number field algorithm analogue applications arithmetic asymptotic cellular automata characteristic polynomial characteristic roots co-prime coefficients computation congruence conjecture considered construction continued fraction defined degree discrepancy elements elliptic curve elliptic divisibility sequences estimate example exponential functions exponential polynomials exponential sums Fibonacci finite fields fixed prime given gives growth rate Hadamard implies improved initial values integer ideals irreducible known Lehmer Let a denote linear complexity linear forms linear recurrence sequences log log logarithms lower bound Lucas sequences Math matrix Mersenne multiplicative non-degenerate linear recurrence non-linear recurrence sequences non-trivial non-zero number of solutions number of zeros number theory obtained p-adic papers particular polynomial F power series prime divisors prime ideal prime numbers problem properties prove pseudo-random number quadratic quences question quotients rational function realizable recurrence relation residue ring root of unity S-unit Section sequence of order sequences satisfying shows studied upper bound