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 more special than G-functions.

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.

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 Shidlovsky and Shidlovskii theorem), named after Carl Ludwig Siegel and Andrei Borisovich Shidlovskii.

Suppose that we are given n E-functions, E₁(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 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

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.

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