Transcendental Numbers

This book is devoted to one of the directions of research in the theory of transcen-
dental numbers. It includes an exposition of the fundamental results concerning
the arithmetic properties of the values of E-functions which satisfy linear differ-
ential equations with coefficients in the field of rational functions.
The notion of an E-function was introduced in 1929 by Siegel, who created
a method of proving transcendence and algebraic independence of the values of
such functions. An E-function is an entire function whose Taylor series coeffi-
cients with respect to z are algebraic numbers with certain arithmetic properties.
The simplest example of a transcendental E-function is the exponential function
e Z . In some sense Siegel's method is a generalization of the classical Hermite-
Lindemann method for proving the transcendence of e and 1f and obtaining some
other results about arithmetic properties of values of the exponential function at
algebraic points.
In the course of the past 30 years, Siegel's method has been further developed
and generalized. Many papers have appeared with general theorems on transcen-
dence and algebraic independence of values of E-functions; estimates have been
obtained for measures of linear independence, transcendence and algebraic inde-
pendence of such values; and the general theorems have been applied to various
classes of concrete E-functions. The need naturally arose for a monograph bring-
ing together the most fundamental of these results. The present book is an attempt
to meet this need.