# E-function

In mathematics, **E-functions** are a type of power series that satisfy particular arithmetic conditions on the coefficients. They are of interest in transcendental number theory, and are more special than G-functions.

## Definition

A function *f*(*x*) is called of **type E**, or an

**,[1] if the power series**

*E*-functionsatisfies the following three conditions:

- All the coefficients
*c*belong to the same algebraic number field,_{n}*K*, which has finite degree over the rational numbers; - For all ε > 0,

- ,

where the left hand side represents the maximum of the absolute values of all the algebraic conjugates of *c _{n}*;

- For all ε > 0 there is a sequence of natural numbers
*q*_{0},*q*_{1},*q*_{2},... such that*q*is an algebraic integer in_{n}c_{k}*K*for*k*=0, 1, 2,...,*n*, and*n*= 0, 1, 2,... and for which

- .

The second condition implies that *f* is an entire function of *x*.

## Uses

*E*-functions were first studied by Siegel in 1929.[2] He found a method to show that the values taken by certain *E*-functions were algebraically independent.This was a result which established the algebraic independence of classes of numbers rather than just linear independence.[3] Since then these functions have proved somewhat useful in number theory and in particular they have application in transcendence proofs and differential equations.[4]

## The Siegel–Shidlovsky theorem

Perhaps the main result connected to *E*-functions is the Siegel–Shidlovsky theorem (also known as the Shidlovsky and Shidlovskii theorem), named after Carl Ludwig Siegel and Andrei Borisovich Shidlovskii.

Suppose that we are given *n* *E*-functions, *E*_{1}(*x*),...,*E*_{n}(*x*), that satisfy a system of homogeneous linear differential equations

where the *f _{ij}* are rational functions of

*x*, and the coefficients of each

*E*and

*f*are elements of an algebraic number field

*K*. Then the theorem states that if

*E*

_{1}(

*x*),...,

*E*

_{n}(

*x*) are algebraically independent over

*K*(

*x*), then for any non-zero algebraic number α that is not a pole of any of the

*f*the numbers

_{ij}*E*

_{1}(α),...,

*E*

_{n}(α) are algebraically independent.

## Examples

- Any polynomial with algebraic coefficients is a simple example of an
*E*-function. - The exponential function is an
*E*-function, in its case*c*=1 for all of the_{n}*n*. - If λ is an algebraic number then the Bessel function
*J*_{λ}is an*E*-function. - The sum or product of two
*E*-functions is an*E*-function. In particular*E*-functions form a ring. - If
*a*is an algebraic number and*f*(*x*) is an*E*-function then*f*(*ax*) will be an*E*-function. - If
*f*(*x*) is an*E*-function then the derivative and integral of*f*are also*E*-functions.

## References

- Carl Ludwig Siegel,
*Transcendental Numbers*, p.33, Princeton University Press, 1949. - C.L. Siegel,
*Über einige Anwendungen diophantischer Approximationen*, Abh. Preuss. Akad. Wiss.**1**, 1929. - Alan Baker,
*Transcendental Number Theory*, pp.109-112, Cambridge University Press, 1975. - Serge Lang,
*Introduction to Transcendental Numbers*, pp.76-77, Addison-Wesley Publishing Company, 1966.