In surveys such as this, it is conventional to attach a list of open questions.
Rather than doing this, the best current results known to the authors are pre-
sented; if a generalization is straightforward and can be done in the framework of
the same arguments that is noted. Other generalizations or improvements should
be considered implicit research problems. We do however mention attempts at
improvements which seem hopeless in the light of today's knowledge.
Finally, we add several words about what we do not deal with. First, it is strik-
ing to note that the binary recurrence u(n + 2) = u(n + l)+u(n), one of the simplest
linear recurrences whose solutions are not geometric progressions, has been a sub-
ject of mathematical scrutiny certainly since the publication of Leonardo of Pisa's
Liber abaci in 1202 [1212]. Indeed, this recurrence has an entire journal devoted to
it [113]. This volume is more egalitarian; with a few exceptions, no special prop-
erties of individual recurrences will be discussed. Several specific sequences arise
as examples; the most important of these are listed with their identifying numbers
in Sloane's Online Encyclopedia of Integer Sequences [1222] in an Appendix on
page 254.
Second, one could write an enormous book devoted to one particular case of
linear recurrence sequences polynomials. We do not deal with polynomials per
se; extensive treatments are in [1116] and [1120]. Nonetheless, this case alone
justifies the great interest in general linear recurrence sequences. Therefore, we
give several applications to polynomials but such applications are obtained using
partially hidden although not too deep links between polynomials and linear
recurrence sequences.
Third, a huge book could be written dealing with exponential polynomials as
examples of entire functions and therefore, ultimately, with analytic properties of
those functions. We barely consider any analytical features of exponential poly-
nomials, though we mention some relevant results about the distribution of their
zeros. We do not deal with analytical properties of iteration of polynomial map-
pings. Thus the general field of complex dynamics, and the celebrated Mandelbrot
set, is outside our scope. (Recall that the Mandelbrot set is the set of points c G C
for which the sequence of polynomial iterations z(k) = z(k l) 2 + c, z(0) = 0, is
bounded; for details we refer to [154].) However, in Chapter 3 we do consider some
simple periodic properties of this and more general mappings.
Fourth, as we mentioned, general statements about the behaviour both
Archimedean and non-Archimedean of sums of 5-units lie in the background
of important results on linear recurrence sequences. Nonetheless, we do not deal
with sums of S-units or their applications systematically. On the topic generally,
we first recommend the pioneering papers [376] and [1037] which appeared inde-
pendently and contemporaneously (the latter as a preprint [1019] of Macquarie
University in 1982). We point particularly to the book [1181] and the excellent
survey papers [378], [380], [381], [382], [503], [1128], [1175], [1285], [1286].
On the other hand, we do present some less well-known results about finitely
generated groups, such as estimates of the size of their reduction modulo an integer
ideal in an algebraic number field, and on the testing of multiplicative independence
of their generators. When results on 5-unit sums are applied to linear recurrence
sequences, an induction argument usually allows the conditions on non-vanishing
proper sub-sums to be eliminated (such conditions are unavoidable in the general
study of S-unit sums).
Previous Page Next Page