Introduction
Many sequences of combinatorial interest are known to be unimodal or log-concave and there
has been a considerable amount of interest and research devoted to this topic in recent years.
Even though these two properties have "one-line" definitions it has now become apparent
that to prove the unimodality or log-concavity of a sequence is often a very difficult task that
requires the use of refined and sophisticated mathematical tools. The number and variety
of these tools has been constantly increasing and is quite bewildering and surprising. They
include, for example, classical analysis (see, e.g., [2], [56], [68]), linear algebra (see, e.g., [47]),
the representation theory of Lie algebras and superalgebras (see, e.g., [47], [49], [61], [62]),
algebraic geometry (see, e.g., [65]) as well as the use of bisections and injections (see, e.g.,
[53]). An excellent survey of all these techniques is [65] which also contains many further
references.
The main object of this work is to point out another branch of mathematics that can
be successfully used to attack the kind of problems above referred to, namely, the theory of
total positivity. This theory comes about in unimodality and log-concavity questions in a
quite natural way, in fact, whenever we have in mathematics a set of "objects" that we want
to study we immediately look for "morphisms" between these objects that "preserve" their
structure. In the case at hand (unimodal and log-concave sequences) it is therefore natural
to look for transformations of Rrf into itself that preserve the unimodality or log-concavity
property. Now, the simplest kind of transformations that we can think of are, of course,
the linear transformations. Simple as this idea is it seems that no systematic study of these
kind of linear transformations has ever been made. One possible reason for this is that
examples show that there are very few linear transformations that preserve the unimodal
or log-concave property. However, a closer look reveals that a (very natural) strengthening
of these concepts, that of a Polya frequency sequence (a concept closely connected to that
of total positivity), is much better behaved. The fact that many of the unimodal and log-
concave sequences arising in combinatorics (as well as in algebra, geometry and probability)
turn out to be actually Polya frequency sequences was enough motivation for this author
to begin the study of linear transformations that preserve the Polya frequency (or PF, for
short) property.
This study has turned out to be a very rich subject that can easily fill a mathematician's
life. For this reason, and because the author's interest in these matters stemmed from
combinatorial motivations, only the most fundamental of these linear transformations are
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