Resolution of an integral equation with the Thue–Morse sequence

It is a classical fact that the exponential function is a solution of the integral equation ∫0Xf(x)dx+f(0)=f(X). If we slightly modify this equation to ∫0Xf(x)dx+f(0)=f(αX) with α∈]0,1[α∈]0,1[, it seems that no classical techniques apply to yield solutions. In this article, we consider the parameter α=1/2α=1/2. We will show the existence of a solution which takes the values of the Thue–Morse sequence on the odd integers.