The Zeros of Faber Polynomials for An M-Cusped Hypocycloid

The Faber polynomials for a region of the complex plane are of interest as a basis for polynomial approximation of analytic functions. In this paper we determine the location, density, and asymptotic behavior of the zeros of Faber polynomials associated with the closed region bounded by the m -cusped hypocycloid with parametric equation z = exp(iθ) + 1(m − 1)exp(−(m − 1)iθ), 0≤θ<2π, m 2,3,4,... . For m = 2, the Faber polynomials are simply the classical Chebyshev polynomials for the segment [−2,2]; thus our results can be viewed as a study of the algebraic and asymptotic properties of generalized Chebyshev polynomials.