E-function

In mathematics, "E"-functions are a type of power series that satisfy particular systems of linear differential equations.

Definition

A function "f"("x") is called of type "E", or an "E"-function [Carl Ludwig Siegel, "Transcendental Numbers", p.33, Princeton University Press, 1949.] , 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,:$overline\{left|c\_\{n\}\; ight=Oleft(n^\{nvarepsilon\}\; ight)$,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 "q"₀, "q"₁, "q"₂,… 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:$q\_\{n\}=Oleft(n^\{nvarepsilon\}\; ight)$.

The second condition implies that "f" is an entire function of "x".

Uses

"E"-functions were first studied by Siegel in 1929 [C.L. Siegel, "Über einige Anwendungen diophantischer Approximationen", Abh. Preuss. Akad. Wiss. 1, 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 [Alan Baker, "Transcendental Number Theory", pp.109-112, Cambridge University Press, 1975.] . Since then these functions have proved somewhat useful in number theory and in particular they have application in transcendence proofs and differential equations [Serge Lang, "Introduction to Transcendental Numbers", pp.76-77, Addison-Wesley Publishing Company, 1966.] .

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, "E"₁("x"),…,"E"_"n"("x"), that satisfy a system of homogeneous linear differential equations:$y^prime\_i=sum\_\{j=1\}^n\; f\_\{ij\}(x)y\_jquad(1leq\; ileq\; n)$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 "E"₁("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 "E"₁(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

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