Remark on the Distribution of Chebychev Polynomials on [−1, 1]

Let (T n(x))n≥0 denote the sequences of Chebychev polynomials of the first kind defined by the recursive relation T 0(x) ≡ 1, T 1(x) = xand T n+1(x) = 2xT n(x) − T n−1(x). We observe that the sequences T n(x) n≥0 is asymptotically distributed with respect to the measure with the density π−1(1 − x2) − 1 2 on [−1, 1] for almost all x. Various refinements of this observation are also noted.