## An Introduction to Difference EquationsThe second edition has greatly benefited from a sizable number of comments and suggestions I received from users of the book. I hope that I have corrected all the er rors and misprints in the book. Important revisions were made in Chapters I and 4. In Chapter I, we added two appendices (global stability and periodic solutions). In Chapter 4, we added a section on applications to mathematical biology. Influenced by a friendly and some not so friendly comments about Chapter 8 (previously Chapter 7: Asymptotic Behavior of Difference Equations), I rewrote the chapter with additional material on Birkhoff's theory. Also, due to popular demand, a new chapter (Chapter 9) under the title "Applications to Continued Fractions and Orthogonal Polynomials" has been added. This chapter gives a rather thorough presentation of continued fractions and orthogonal polynomials and their intimate connection to second-order difference equations. Chapter 8 (Oscillation Theory) has now become Chapter 7. Accordingly, the new revised suggestions for using the text are as follows. The diagram on p. viii shows the interdependence of the chapters The book may be used with considerable flexibility. For a one-semester course, one may choose one of the following options: (i) If you want a course that emphasizes stability and control, then you may select Chapters I, 2, 3, and parts of 4, 5, and 6. This is perhaps appropriate for a class populated by mathematics, physics, and engineering majors. |

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### Contents

Systems of Difference Equations | 105 |

Stability Theory | 154 |

The ZTransform Method | 215 |

Copyright | |

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analysis apply assume assumption asymptotically stable Ax(n behavior bounded called Chapter characteristic equation characteristic roots completely completely controllable completely observable compute conclude condition Consequently Consider constant continued fraction controllable converges Corollary corresponding defined definite determine difference equation discrete eigenvalues eigenvectors equation x(n equilibrium point Example Exercises exists FIGURE formula function fundamental matrix given gives Hence hold implies initial integer introduce j=no Jordan Lemma Liapunov limit linear linearly independent Math Mathematical method multiplicity nonsingular Notice observable obtain origin oscillates pair period pi(n Pn(x positive probability Problem produces Proof Prove radius of convergence rank reader Remark result satisfies sequence Show simple solution x(n Solve Suppose symmetric matrix system x(n Theorem theory transform uniformly unit unstable vector write xi(n yields yı(n York Z-transform zero solution