Saber Elaydi suggests in his Preface to the third edition of An Introduction to Difference Equations that this textbook for “advanced undergraduate and beginning graduate [students]” be used for a two-semester course, following one of four options as a function of taste and interest: stability theory, asymptotic theory, oscillation theory, or control theory. The book seems to fit the bill perfectly. It is very well-written and thorough in its coverage of topics (which are fascinating and numerous, notwithstanding the author’s insistence that he is not interested in encyclopedic coverage). Additionally the book is full of good exercises at all levels replete with hints and answers (although Elaydi even goes so far at to sprinkle in an occasional unsolved problem!) and is rich in good examples. It is impossible not to admire Elaydi’s achievement in putting together a textbook of such quality.
Difference equations are of course to some extent analogous to differential equations and this parallel is already evident in the familiar appearance of the general solution of a homogeneous linear difference equation with constant coefficients: just as with homogeneous linear ordinary differential equations (also with constant coefficients, of course) one immediately goes over to an associated characteristic polynomial whose roots are intimately involved in the manufacture of the according general solution. In this connection Elaydi presents e.g. the example of the difference equation realizing the Fibonacci numbers, the upshot being that, qua general solution, the general Fibonacci number is expressible in terms of powers of the golden section and its conjugate, the roots of one of the most famous characteristic polynomials of all.
Proceeding to the non-homogenous case we encounter further parallels with ordinary differential equations: the methods of undetermined coefficients (which is actually misspelled as “undetermind coefficeints” — an isolated singularity) and variation of parameters appear, soon to be followed by a study of the limiting behavior of solutions. Elaydi goes on to play with some nice examples, featuring an elegant treatment of the gambler’s ruin problem.
There is of course a thorough treatment of difference equations per se, culminating in a discussion of Markov chains. Then we encounter stability theory, including coverage of Lyupanov’s second method, and a discussion of Z-transforms and Volterra difference equations featuring a section titled, “The Z-transform Versus the Laplace Transform.”
After a chapter on oscillation theory, Elaydi hones in on the asymptotic behavior of difference equations in greater detail, with the focus falling on Poincaré’s Theorem and its extension by O. Perron. This is beautiful mathematics: even in the case of homogeneous linear difference equations with possibly non-constant coefficients, the general solution, x(n), generally has the property that the quotient x(n+1)/x(n) tends (as n → ∞) to a root of an associated (still “characteristic”) polynomial, with the case of constant coefficients a leitmotiv (as exemplified dramatically by, again, the Fibonacci numbers with F(n+1)/F(n)→ λ, the golden section).
Subsequently the theory of continued fractions is given a systematic going-over and Elaydi’s treatment of the connection between continued fractions and infinite series is particularly evocative: it even includes some very nice material on the Riemann ζ-function (if k is an integer ≥2, ζ(k) itself is rendered as a continued fraction with wonderfully suggestive numerators and denominators).
Thus, An Introduction to Difference Equations is a terrific book almost every page of which contains marvelous things. It will serve all the pedagogical purposes Elaydi delineates, even though the wealth of material in the book will often tempt the reader to go off in tangent or orthogonal directions at the risk of destroying the pace of the coverage. But wherever one ends up, it will be a trip well worth taking.
Michael Berg is Professor of Mathematics at Loyola Marymount University in California.