On a distribution function of a probability measure involving a permutation

Abstract

In [3], we have introduced a probability measure to study the power and exponential sums for a certain coding system. The distribution function of the probability measure gives explicit formulas for the power and exponential sums. [3,Theorem 4] states that the higher order derivatives of the distribution function with respect to a certain parameter are expressed by a generalization of the Takagi function. In [3], we only gave the sketch of the proof of Theorem 4, because the complete proof is very long. The purpose of this paper is to give the complete proof of [3,Theorem 4].