E-function

Template:For 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 closely related to G-functions.

Definition

A power series with coefficients in the field of algebraic numbers

$f(x)=\sum _{n=0}^{\mathrm{\infty }}{c}_n\frac{x^n}{n!}\in \bar{\mathbb{Q}}[[x]]$

is called an Template:Math-function if it satisfies the following three conditions:

• It is a solution of a non-zero linear differential equation with polynomial coefficients (this implies that all the coefficients Template:Math belong to the same algebraic number field, Template:Math, which has finite degree over the rational numbers);
• For all $\epsilon >0$,   $\overline{\left|c_n\right|}=O\left(n^{n\epsilon }\right),$

where the left hand side represents the maximum of the absolute values of all the algebraic conjugates of Template:Math;

• For all $\epsilon >0$ there is a sequence of natural numbers Template:Math such that Template:Math is an algebraic integer in Template:Math for Template:Math, and Template:Math and for which $q_n=O\left(n^{n\epsilon }\right).$

The second condition implies that Template:Math is an entire function of Template:Math.

Uses

Template:Math-functions were first studied by Siegel in 1929. He found a method to show that the values taken by certain Template:Math-functions were algebraically independent. 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 and in particular they have application in transcendence proofs and differential equations.

The Siegel–Shidlovsky theorem

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

Suppose that we are given Template:Math Template:Math-functions, Template:Math, that satisfy a system of homogeneous linear differential equations

${y_i}^{\prime }=\sum _{j=1}^n{f}_{ij}(x)y_j(1\le i\le n)$

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

Examples

1. Any polynomial with algebraic coefficients is a simple example of an Template:Math-function.
2. The exponential function is an Template:Math-function, in its case Template:Math for all of the Template:Math.
3. If Template:Math is an algebraic number then the Bessel function Template:Math is an Template:Math-function.
4. The sum or product of two Template:Math-functions is an Template:Math-function. In particular Template:Math-functions form a ring.
5. If Template:Math is an algebraic number and Template:Math is an Template:Math-function then Template:Math will be an Template:Math-function.
6. If Template:Math is an Template:Math-function then the derivative and integral of Template:Math are also Template:Math-functions.

References

• Template:Mathworld