Resolution of an integral equation with the Thue-Morse sequence

TL;DR

This paper investigates a modified integral equation involving a parameter and demonstrates the existence of a solution that aligns with the Thue-Morse sequence at odd integers, revealing new connections between integral equations and combinatorial sequences.

Contribution

It introduces a novel integral equation with a scaled argument and proves the existence of solutions related to the Thue-Morse sequence, a connection not previously established.

Findings

Existence of solutions matching the Thue-Morse sequence on odd integers

Extension of classical integral equations with a scaled argument

New link between integral equations and combinatorial sequences

Abstract

It is a classical fact that the exponential function is solution of the integral equation $ \int_0^X f(x)dx + f(0) =f(X)$. If we slightly modify this equation to $ \int_0^X f(x)dx+f(0)=f(\alpha X)$ with $\alpha\in ]0,1[$, it seems that no classical techniques apply to yields solutions. In this article, we consider the parameter $\alpha=1/2$. We will show the existence of a solution wich takes the values of the Thue-Morse sequence on the odd integers.