mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, E-functions are a type of

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...

that satisfy particular arithmetic conditions on the coefficients. They are of interest in

transcendental number theory Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways. Transcendence ...

, and are more special than G-functions.

Definition

A function is called of type , or an -function, if the power series :

f(x)=\sum_^\infty c_n \frac

satisfies the following three conditions: * All the coefficients belong to the same

algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...

, , which has finite degree over the rational numbers; * For all

\varepsilon>0

\overline=O\left(n^\right),

: where the left hand side represents the maximum of the absolute values of all the algebraic conjugates of ; * For all

\varepsilon>0

there is a sequence of natural numbers such that is an

algebraic integer In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...

in for , and and for which

q_n=O\left(n^\right).

The second condition implies that is an

entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...

of .

Uses

-functions were first studied by

Siegel Siegel (also Segal or Segel), is a German and Ashkenazi Jewish surname. it can be traced to 11th century Bavaria and was used by people who made wax seals for or sealed official documents (each such male being described as a ''Siegelbeamter''). ...

in 1929. He found a method to show that the values taken by certain -functions were

algebraically independent In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K. In particular, a one element set \ is algebraically in ...

. This was a result which established the algebraic independence of classes of numbers rather than just linear independence. Since then these functions have proved somewhat useful in

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

and in particular they have application in transcendence proofs and

differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

. Serge Lang, ''Introduction to Transcendental Numbers'', pp.76-77, Addison-Wesley Publishing Company, 1966.

The Siegel–Shidlovsky theorem

Perhaps the main result connected to -functions is the Siegel–Shidlovsky theorem (also known as the Siegel and Shidlovsky theorem), named after

Carl Ludwig Siegel Carl Ludwig Siegel (31 December 1896 – 4 April 1981) was a German mathematician specialising in analytic number theory. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation, ...

and Andrei Borisovich Shidlovsky. Suppose that we are given -functions, , that satisfy a system of homogeneous linear differential equations :

y^\prime_i=\sum_^n f_(x)y_j\quad(1\leq i\leq n)

where the are rational functions of , and the coefficients of each and are elements of an algebraic number field . Then the theorem states that if are algebraically independent over , then for any non-zero algebraic number that is not a pole of any of the the numbers are algebraically independent.

Examples

# Any polynomial with algebraic coefficients is a simple example of an -function. # The

exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...

is an -function, in its case for all of the . # If is an algebraic number then the

Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...

is an -function. # The sum or product of two -functions is an -function. In particular -functions form a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

. # If is an algebraic number and is an -function then will be an -function. # If is an -function then the derivative and integral of are also -functions.

References

* {{mathworld, title=E-Function, urlname=E-Function Number theory