# Generalized Power Series on a Non-Archimedean Field

### Abstract

Power series with rational exponents on the real number field and the
Levi-Civita field are studied. We derive a radius of convergence for power
series with rational exponents over the field of real numbers that depends on
the coefficients and on the density of the exponents in the series. Then we
generalize that result and study power series with rational exponents on the
Levi-Civita field. A radius of convergence is established that asserts
convergence under a weak topology and reduces to the conventional radius of
convergence for real power series. It also asserts strong (order) convergence
for points whose distance from the center is infinitely smaller than the
radius of convergence. Then we study a class of functions that are given
locally by power series with rational exponents, which are shown to form a
commutative algebra over the Levi-Civita field; and we study the
differentiability properties of such functions within their domain of
convergence.

K. Shamseddine, M. Berz,
*Indagationes Mathematicae* **17,3** (2006) 457-477

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