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.
A function f(x) is called of type E, or an E-function, [1] if the power series
satisfies the following three conditions:
The second condition implies that f is an entire function of x.
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]
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),...,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 and f are elements of an algebraic number field K. Then the theorem states that if E1(x),...,En(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 E1(α),...,En(α) are algebraically independent.
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.
A function f(x) is called of type E, or an E-function, [1] if the power series
satisfies the following three conditions:
The second condition implies that f is an entire function of x.
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]
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),...,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 and f are elements of an algebraic number field K. Then the theorem states that if E1(x),...,En(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 E1(α),...,En(α) are algebraically independent.