E-function

For the generalization of hypergeometric series, see MacRobert 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

${\displaystyle f(x)=\sum _{n=0}^{\infty }c_{n}{\frac {x^{n}}{n!}}\in {\overline {\mathbb {Q} }}[\![x]\!]}$

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 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 ${\displaystyle \varepsilon >0}$,   ${\displaystyle {\overline {\left|c_{n}\right|}}=O\left(n^{n\varepsilon }\right),}$

where the left hand side represents the maximum of the absolute values of all the algebraic conjugates of c_n;

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

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

${\displaystyle y_{i}^{\prime }=\sum _{j=1}^{n}f_{ij}(x)y_{j}\quad (1\leq i\leq n)}$

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 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.

Examples

1. Any polynomial with algebraic coefficients is a simple example of an E-function.
2. The exponential function is an E-function, in its case c_n = 1 for all of the n.
3. If λ is an algebraic number then the Bessel function J_λ is an E-function.
4. The sum or product of two E-functions is an E-function. In particular E-functions form a ring.
5. If a is an algebraic number and f(x) is an E-function then f(ax) will be an E-function.
6. If f(x) is an E-function then the derivative and integral of f are also E-functions.