Continuous Functions on Discrete Valuation Rings

LetRbe a complete discrete valuation ring with finite residue field, letKbe its quotient field. We construct polynomial functionsϕ(n,a)(n=0,1,…) such that any continuous functionffromRintoKhas the following expansionf(a)=∑n=0∞anϕ(n,a)where the sequence {an}⊂Kis uniquely determined byfand satisfies thatlimn→∞an=0. WhenK=Qp, if we replaceϕ(n,=a) by the binomial coefficienta(a−1)…(a−n+1)/n! we have Mahler's expansion theorem.