Chebyshev polynomials and nested square roots

The purpose of this note is to report a curious relation between the Chebyshev polynomials of degree 2n in a complex variable and the nested square roots of depth n of the form ±2±2±⋯±2+2ξ,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,∞).