Abstract. In this paper, we give an \emph {explicit} representation of the complex Chebyshev polynomials on a given arc of the unit circle (in the complex plane) in terms of real Chebyshev polynomials on two symmetric intervals (on the real line). The real Chebyshev polynomials, for their part, can be expressed via a conformal mapping with the help of Jacobian elliptic and theta functions, which goes back to the work of Akhiezer in the 1930's.

DOI: 10.14232/actasm-018-343-y

AMS Subject Classification (1991): 30E10, 30C10, 33E05, 41A50

Keyword(s): Chebyshev polynomials, circular arc, Jacobian elliptic function, Jacobian theta function

Received October 29, 2018 and in final form January 16, 2019. (Registered under 93/2018.)