-functions are a type of power series
that satisfy particular systems of linear differential equations.
A function f(x) is called of type E, or an E-function, if the power series
satisfies the following three conditions:
where the left hand side represents the maximum of the absolute values of all the algebraic conjugates
- For all ε > 0 there is a sequence of natural numbers q0, q1, q2,… such that qnck is an algebraic integer in K for k=0, 1, 2,…, n, and n = 0, 1, 2,… and for which
The second condition implies that f is an entire function of x.
E-functions were first studied by Siegel in 1929. 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. Since then these functions have proved somewhat useful in number theory and in particular they have application in transcendence proofs and differential equations.
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),…,En(x), that satisfy a system of homogeneous linear differential equations
where the fij
are rational functions of x
, and the coefficients of each E
are elements of an algebraic number field K
. Then the theorem states that if E1
) are algebraically independent over K
), then for any non-zero algebraic number α that is not a pole of any of the fij
the numbers E1
(α) are algebraically independent.
- Any polynomial with algebraic coefficients is a simple example of an E-function.
- The exponential function is an E-function, in its case cn=1 for all of the n.
- If λ is an algebraic number then the Bessel function Jλ 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.