The limit of a function, this fundamental concept that began to take shape from the time of Archimedes, is the central concept around which the field of Mathematical Analysis develops. It is well known to educators the difficulties students face in assimilating it to a satisfactory degree in order to understand the basic concepts of Differential and Integral Calculus and Differential Equations. Difference Equations, which achieve results analogous to those of Differential Equations, have the great advantage that their presentation does not presuppose the knowledge of the complex concept of a limit. Difference equations are developed in a simple and understandable way and do not require knowledge of specific mathematical concepts, except, of course, the elementary ones. For this reason, many mathematicians today believe that their teaching can begin even in Secondary Education.
It is particularly noted that the importance of Difference Equations is mainly due to the rapid development of Electronic Computers, which use mathematical models with exclusively discrete variables and not continuous ones.
It should also be noted that both Greek and foreign literature lack a comprehensive presentation of topics on Difference Equations. This book is expected to cover a significant part of this gap.
The book is divided into two parts. The first and most important part contains the theory of Difference Equations and the second the Special Functions for the derivation of which Difference Equations are used. At the end of each independent section, there is a summary of the theory for revision and consolidation of the concepts mentioned and a significant number of unsolved exercises with their answers.
In the structuring of the material, emphasis was given not to detailed theoretical proofs, but to the solution of many selected representative examples and applications, such as, for example, in Automatic Control and the Chaotic Theory of Discrete Systems.

Contents

Part One: Difference Equations
Chapter I. Linear difference equations

 Introduction
    Definitions. Existence
    First-order linear difference equations
    Applications
    Higher-order linear difference equations
    Homogeneous equations with constant coefficients
    Non-homogeneous equations with constant coefficients
    Applications

Chapter II. Linear equations with non-constant coefficients

    The method of variation of parameters
    Reduction of the order of a linear equation
    Euler's difference equation

Chapter III. Non-linear difference equations

    Special examples of non-linear difference equations
    Functional difference equations
    A characteristic value problem
    Approximating differential equations with difference equations

Chapter IV. Systems of linear difference equations

    Systems of linear difference equations with constant coefficients
    Non-homogeneous linear systems
    Z-transform
    Non-autonomous systems of difference equations

Chapter V. General Applications of Difference Equations

    Introduction
    Controlled Systems
    Observable Systems
    Stability of Difference Equations
    Stability of Systems of Linear Difference Equations
    Difference Equations and Chaos

Part Two: Special Functions
Chapter I. Special Functions

    Functions Defined by Generalized Integrals
    Gamma Function
    Beta Function
    Functions Defined by Solutions of Differential Equations
    Generating Function
    Legendre Functions
    The Sturm-Liouville Problem