E-function

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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 cn belong to the same algebraic number field, K, which has finite degree over the rational field \mathbb{Q},
  2. For all ε > 0, \overline{\left|c_{n}\right|}=O\left(n^{n\epsilon}\right) as n\rightarrow\infty,
  3. For all ε > 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 cn.

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.

[edit] 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 cn = 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.

[edit] References

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