On Faber polynomials generated by an $m$-star

Abstract:

In this paper, we study the Faber polynomials ${F_n}(z)$ generated by a regular m-star $(m = 3,4, \ldots )$ \[ {S_m} = \{ {x{\omega ^k};0 \leq x \leq {4^{1/m}},k = 0,1, \ldots ,m - 1,{\omega ^m} = 1} \}.\] An explicit and precise expression for ${F_n}(z)$ is obtained by computing the coefficients via a Cauchy integral formula. The location and limiting distribution of zeros of ${F_n}(z)$ are explored. We also find a class of second-order hypergeometric differential equations satisfied by ${F_n}(z)$. Our results extend some classical results of Chebyshev polynomials for a segment $[ - 2,2]$ in the case when $m = 2$.