The Faber polynomials for circular lunes

We study the Faber polynomials Fn(z) generated by a circular lune symmetric about both axes with vertices at the points z = ±α (0 < α ≤ 2) and exterior angle απ. An explicit expression of Fn(z) was obtained by computing the coefficients via a Cauchy integral formula. We also illustrate the location of the zeros of Faber polynomial and of its derivative. Our results include a circle and a segment as special cases when α = 1, 2, respectively.