1.1. MAIN DEFINITIONS AND PRINCIPAL PROPERTIE S 7

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

m

(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

s

y£Pj(X)rtfX)

= 0.

These and similar functions satisfying

s

J2Pj(X)R(X«J)

= 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].