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We give a simple explicit formula of the exponential sum of digital sums by use of the distribution function of the binomial moasure. We also apply our formula to the study of the power sums of digital sums.

We state an expansion theorem of continuous functions based on the multinomial measure, which is a generalization of the Schauder one, and then apply it to the study of a system of infinitely many difference equations. We also study the differentiability of the distribution function of the multinomial measure with respect to the parameters needed to define the multinomial measures.