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 closely related to G-functions.

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 with polynomial coefficients (this is implies that all the coefficients $c n$ belong to the same algebraic number field, $K$ , which has finite degree 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 of

c n

;

For all $\varepsilon >0$ 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 $q_{n}=O\left(n^{n\varepsilon }\right).$

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 Siegel and Shidlovsky theorem), named after 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_{i}^{\prime }=\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 $E 1 (x),..., E n (x)$ 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

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.

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.

Weisstein, Eric W. "E-Function". MathWorld.

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

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

[3] Alan Baker, Transcendental Number Theory, pp.109-112, Cambridge University Press, 1975.

[4] Serge Lang, Introduction to Transcendental Numbers, pp.76-77, Addison-Wesley Publishing Company, 1966.

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