Chebyshev polynomials and nested square roots

The purpose of this note is to report a curious relation between the Chebyshev polynomials of degree 2(n) in a complex variable and the nested square roots of depth n of the form +/-root 2 +/-root 2 +/- ... +/- root 2 + 2 xi, xi being a complex number. Our approach leads us to generalize and recover, in an unified manner, the closed-form expressions recently given by Servi, corresponding to the case xi is an element of [-1, 1], and by Nyblom, corresponding to xi left perpendicular1, infinity). (C) 2012 Elsevier Inc. All rights reserved.