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
- [math]\displaystyle{ f(x)=\sum_{n=0}^\infty c_n \frac{x^n}{n!} }[/math]
satisfies the following three conditions:
- All the coefficients cn belong to the same algebraic number field, K, which has finite degree over the rational numbers;
- For all [math]\displaystyle{ \varepsilon\gt 0 }[/math], [math]\displaystyle{ \overline{\left|c_n\right|}=O\left(n^{n\varepsilon}\right), }[/math]
- where the left hand side represents the maximum of the absolute values of all the algebraic conjugates of cn;
- For all [math]\displaystyle{ \varepsilon\gt 0 }[/math] 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 [math]\displaystyle{ q_n=O\left(n^{n\varepsilon}\right). }[/math]
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),...,En(x), that satisfy a system of homogeneous linear differential equations
- [math]\displaystyle{ y^\prime_i=\sum_{j=1}^n f_{ij}(x)y_j\quad(1\leq i\leq n) }[/math]
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.
Examples
- 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.
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.
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