Mariconda C., Tonolo A. Discrete Calculus: Methods for Counting

New York: Springer, 2016. - 674 p.This book provides an introduction to combinatorics, finite calculus, formal series, recurrences, and approximations of sums. Readers will find not only coverage of the basic elements of the subjects but also deep insights into a range of less common topics rarely considered within a single book, such as counting with occupancy constraints, a clear distinction between algebraic and analytical properties of formal power series, an introduction to discrete dynamical systems with a thorough description of Sarkovskii’s theorem, symbolic calculus, and a complete description of the Euler-Maclaurin formulas and their applications. Although several books touch on one or more of these aspects, precious few cover all of them. The authors, both pure mathematicians, have attempted to develop methods that will allow the student to formulate a given problem in a precise mathematical framework. The aim is to equip readers with a sound strategy for classifying and solving problems by pursuing a mathematically rigorous yet user-friendly approach. This is particularly useful in combinatorics, a field where, all too often, exercises are solved by means of ad hoc tricks. The book contains more than 400 examples and about 300 problems, and the reader will be able to find the proof of every result. To further assist students and teachers, important matters and comments are highlighted, and parts that can be omitted, at least during a first and perhaps second reading, are identified.Let’s Learn to Count
Counting Sequences and Collections
Occupancy Constraints
Inclusion/Exclusion
Stirling Numbers and Eulerian Numbers
Manipulation of Sums
Formal Power Series
Generating Formal Series and Applications
Recurrence Relations
Linear Recurrence Relations
Symbolic Calculus
The Euler–Maclaurin Formulas of Order 1 and 2
The Euler–Maclaurin Formula of Arbitrary Order
Cauchy and Riemann Sums, Factorials, Ramanujan Numbers and Their Approximations