Cn-Functions over completions of at finite places of

It is known from Wagnerʹs theorem (Acta Arith. 17 (1971) 389) that a continuous function f from to K can be expanded using Carlitz polynomials Gj(x): f(x)=∑j=0∞ aj Gj(x), with equal to the completion of at a prime and K the quotient field of . Using the definition of a Cn-function by Schikhof (Ultra-Metric Calculus: An Introduction to p-adic Analysis, Cambridge Studies in Advanced Mathematics, Vol. 4, Cambridge University Press, Cambridge, 1984), we prove in this paper that f(x) is a Cn-function if and only if limj→∞ aj jn=0. An alternative approach to Wagnerʹs result is also given in the paper.