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 ± 2 ± 2 ± ⋯ ± 2 + 2 ξ , ξ 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 ξ ∈ [ − 1 , 1 ] , and by Nyblom, corresponding to ξ ∈ [ 1 , ∞ ) .