mathematics of computation volume 62.number 205 january 1994, pages 277-287 ON FABER POLYNOMIALS GENERATED BY AN w-STAR J. BARTOLOMEO AND MATTHEW HE Abstract. In this paper, we study the Faber polynomials F„(z) generated by a regular m-star (m = 3, 4, ... ) Sm = {xcok; 0 < x < 41/m, fc = 0,l,...,iw-l, wm = 1}. An explicit and precise expression for F„(z) is obtained by computing the co- efficients via a Cauchy integral formula. The location and limiting distribution of zeros of F„(z) are explored. We also find a class of second-order hyper- geometric differential equations satisfied by Fn(z). Our results extend some classical results of Chebyshev polynomials for a segment [-2, 2] in the case when m = 2 . 1. Introduction Let E be a compact set (not a single point) whose complement C*\E with respect to the extended plane is simply connected. The Riemann mapping the- orem asserts that there exists a conformai mapping w = O(z) of C*\E onto the exterior of a circle \w\ = pe in the u>-plane. For a unique choice of pe , we can insist that <P(oo) = oo, O'(oo) = 1, so that, in a neighborhood of infinity, (1.1) O(z) = z + a0 + ^ + ^ + ---. z zz The polynomial part of {O(z)}", denoted byF„(z) = z" + ••• , is called the Faber polynomial of degree n generated by the set E. Let (1.2) V(w) = w + bo + - + ^ + --- w w1 be the inverse function of w —<P(z). Thus, *¥(w) maps the domain \w\> pE conformally onto C*\E . Faber [2] proved that 0.3) A-Ê3&- N>*.x«. v ' n=0 The Faber polynomials play an important role in approximation theory and geometric function theory. It can be shown that, under suitable conditions, a Received by the editor March 20, 1992 and, in revised form, November 17, 1992. 1991 Mathematics Subject Classification. Primary 30C15, 41A58. Key words and phrases. Faber polynomials, wi-star, zero distribution. ©1994 American Mathematical Society 0025-5718/94 $1.00+ $.25 per page 277 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use