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In mathematics, E-functions are a type of power series that satisfy particular systems of linear differential equations.
## Definition

_{n};## Uses

## The Siegel-Shidlovsky theorem

_{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_{ij} the numbers E_{1}(α),…,E_{n}(α) are algebraically independent.
## Examples

## References

A function f(x) is called of type E, or an E-function, if the power series

- $f(x)=sum\_\{n=0\}^\{infty\}\; c\_\{n\}frac\{x^\{n\}\}\{n!\}$

satisfies the following three conditions:

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

- $overline\{left|c\_\{n\}right|\}=Oleft(n^\{nvarepsilon\}right)$,

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

- $q\_\{n\}=Oleft(n^\{nvarepsilon\}right)$.

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

E-functions were first studied by Siegel in 1929. He found a method to show that the values taken by certain E-functions were algebraically independent, one of the only results of the early twentieth century 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.

Perhaps the main result connected to E-functions is the Siegel-Shidlovsky theorem (known also as the Shidlovsky and Shidlovskii theorem).

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

- $y^prime\_i=sum\_\{j=1\}^n\; f\_\{ij\}(x)y\_jquad(1leq\; ileq\; n)$

- Any polynomial with algebraic coefficients is a simple example of an E-function.
- The exponential function is an E-function, in its case c
_{n}=1 for all of the 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.

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Last updated on Friday July 11, 2008 at 16:16:39 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday July 11, 2008 at 16:16:39 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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