E-function

In mathematics, a function $f(x)$ is called of type E, or an E-function, if

$f(x)=\sum _{n=0}^{\infty }c_{n}{\frac {x^{n}}{n!}}$

is a power series satisfying the following three conditions:

1. All the coefficients $c_{n}$ belong to the same algebraic number field, K, which has finite degree over the rational field $\mathbb {Q}$,
2. For all $\epsilon >0$, ${\overline {\left|c_{n}\right|}}=O\left(n^{n\epsilon }\right)$ as $n\rightarrow \infty$,
3. For all $\epsilon >0$ there is a sequence of natural numbers $q_{0},q_{1},q_{2},\ldots$ such that $q_{n}c_{k}\in \mathbb {Z} _{K}$ for k=0, 1, 2,..., n, and n = 0, 1, 2,....

Here, ${\overline {\left|c_{n}\right|}}$ denotes the maximum of the absolute values of all the algebraic conjugates of $c_{n}$.

The second condition implies that $f\,$ is an entire function of $x$.

E-functions are useful in number theory and have application in transcendence proofs and differential equations. The Shidlovskii theorem is concerned with the algebraic independence of the values of E-functions at singularities of a system of differential equations.

Examples

1. Any polynomial with algebraic coefficients is a simple example of an E-function.
2. The exponential function is an E-function, in its case $c_{n}=1$ for all of the n.
3. The sum or product of two E-functions is an E-function. In particular E-functions form a ring.
4. If $a$ is an algebraic number and $f(x)$ is an E-function then $f(ax)$ will be an E-function.
5. If $f(x)$ is an E-function then $f^{\prime }(x)$ and $\int _{0}^{x}f(t)\,dt$ are E-functions.

References

• Carl Ludwig Siegel, Transcendental Numbers, p.33, Princeton University Press, 1949.
• Weisstein, Eric W. "E-Function". MathWorld.