  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].
