On a distribution function of a probability measure involving a permutation

TL;DR

This paper provides a complete proof of a theorem relating the derivatives of a distribution function of a probability measure to a generalized Takagi function, which was previously only sketched.

Contribution

It offers the full proof of a key theorem connecting distribution function derivatives to a generalized Takagi function, completing prior incomplete work.

Findings

Explicit formulas for power and exponential sums derived from the distribution function.

Derivatives of the distribution function are expressed via a generalized Takagi function.

The complete proof clarifies the mathematical structure of the distribution function.

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].