Fourier series in solutions of Poincaré and Schröder equations

We consider the quadratic family Qc(z) = z2 + c and the solutions of the Poincaré and Schröder functional equations at three different fixed points of Qc. Comparing these solutions yields Fourier series with quickly decreasing coefficients. The problem is connected with the Karlin–McGregor function derived in population biology.