# E-function

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For the generalization of hypergeometric series, see [MacRobert E function](/source/MacRobert_E_function).

In [mathematics](/source/Mathematics), **E-functions** are a type of [power series](/source/Power_series) that satisfy particular arithmetic conditions on the coefficients. They are of interest in [transcendental number theory](/source/Transcendental_number_theory), and are closely related to [G-functions](/source/G-function_(power_series)).

## Definition

A power series with coefficients in the field of algebraic numbers

- f(x)=\sum_{n=0}^\infty c_n \frac{x^n}{n!} \in \overline{\mathbb{Q}}[\![x]\!]

is called an ***E*-function**[1] if it satisfies the following three conditions:

- It is a solution of a non-zero [linear differential equation](/source/Linear_differential_equation) with polynomial coefficients (this implies that all the coefficients *cn* belong to the same [algebraic number field](/source/Algebraic_number_field), *K*, which has [finite degree](/source/Degree_of_a_field_extension) over the rational numbers);
- For all \varepsilon>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](/source/Conjugate_element_(field_theory)) of *cn*;

- For all \varepsilon>0 there is a sequence of natural numbers *q*0, *q*1, *q*2,... such that *qnck* is an [algebraic integer](/source/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](/source/Entire_function) of *x*.

## Uses

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

## The Siegel–Shidlovsky theorem

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

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_j\quad(1\leq i\leq n)

where the *fij* are [rational functions](/source/Rational_function) 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 *fij* the numbers *E*1(α),...,*E**n*(α) are algebraically independent.

## Examples

1. Any polynomial with algebraic coefficients is a simple example of an *E*-function.
1. The [exponential function](/source/Exponential_function) is an *E*-function, in its case *cn* = 1 for all of the *n*.
1. If λ is an algebraic number then the [Bessel function](/source/Bessel_function) *J*λ is an *E*-function.
1. The sum or product of two *E*-functions is an *E*-function. In particular *E*-functions form a [ring](/source/Ring_(mathematics)).
1. If *a* is an algebraic number and *f*(*x*) is an *E*-function then *f*(*ax*) will be an *E*-function.
1. If *f*(*x*) is an *E*-function then the derivative and integral of *f* are also *E*-functions.

## References

1. Carl Ludwig Siegel, *Transcendental Numbers*, p.33, Princeton University Press, 1949.

1. C.L. Siegel, *Über einige Anwendungen diophantischer Approximationen*, Abh. Preuss. Akad. Wiss. **1**, 1929.

1. Alan Baker, *Transcendental Number Theory*, pp.109-112, Cambridge University Press, 1975.

1. [Serge Lang](/source/Serge_Lang), *Introduction to Transcendental Numbers*, pp.76-77, Addison-Wesley Publishing Company, 1966.

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