1.1.13. Generalized exponential polynomials. Given the representation
of recurrence sequences as generalized power sums from Section 1.1.6, it is natural
to study sequences given by more general functions
(1.6) E{z) = ^2Ai(z)eXp(rli(z)),
Z = l
where, say, polynomials ipi appear in the exponents. Such functions are known as
generalized exponential polynomials or just exponential polynomials.
There is intrinsic interest in the study of the distribution of zeros (and other
values) of such functions, and one might hope that these results would provide some
insight into questions more closely related to the central topic of this book. It is
also natural to consider analogous functions in several variables.
1.1.14. Taylor coefficients of solutions of linear differential equations.
The sequence of coefficients of a power series representing a solution of a linear dif-
ferential equation with polynomial coefficients satisfies a linear recurrence relation
(1.7) a(x 4- n) = si(x)a(x + n 1) + + sn-i(x)a(x + 1) + sn(x)a(x)
with rational function coefficients. For further work on the combinatorial and
arithmetic properties of the sequence of coefficients of such functions and their
generalizations, see [78], [84], [200], [734], [735], [1236].
1.1.15. ^-Generalizations. The g-version of sequences satisfying (1.1) has
been studied. Consider sequences satisfying relations
(1.8) a(x + n) = s1(qx)a(x + n - 1) H h sn-1(qx)a(x + 1) + sn(qx)a(x)
with rational functions Sj. Elements of such sequences are the coefficients of power
series expansions of solutions r of functional equations of the form
= 0.
These and similar functions satisfying
= 0
3 = 1
are also studied in [76], [105], [740], [741], [743], [745]. If log* = t then r(t) =
R(X), but this should not be taken to mean that the properties of power series
coefficients for these two types of functions are the same.
One can go further and consider coefficients of power series expansions of other
interesting functions, such as those satisfying algebraic differential or functional
equations, and multi-variate functions. There are many exciting open problems
and ingenious results in this direction. This topic will not be pursued here; there
are expository papers [736], [1030] and several recent research papers studying
zeros of such functions [737], their arithmetic properties [321], [1147] and their
general structure [76], [734], [735], [738], [740], [741], [743], [745], [812], [1024].
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