E-function

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

1 Definition
2 Uses
3 The Siegel-Shidlovsky theorem
4 Examples
5 References

[edit] Definition

A function $f (x)$ is called of type E, or an E-function^[1], 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,

$\overline{\left|c_{n}\right|}=O\left(n^{n\varepsilon}\right)$ ,

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 q₀, q₁, q₂,… such that q_nc_k is an algebraic integer in K for k=0, 1, 2,…, n, and n = 0, 1, 2,… and for which

$q_{n}=O\left(n^{n\varepsilon}\right)$ .

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

[edit] 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, one of the only results of the early twentieth century 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].

[edit] The Siegel-Shidlovsky theorem

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, E₁(x),…,E_n(x), that satisfy a system of homogeneous linear differential equations

$y^\prime_i=\sum_{j=1}^n f_{ij}(x)y_j\quad(1\leq i\leq n)$

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 E₁(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 E₁(α),…,E_n(α) are algebraically independent.

[edit] 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_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.

[edit] 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.