It is shown that for m = l , 2 , 3 the trigonometric sums ELi(-')* ' c o t^'^f c -\)n/4n) and 5Z*=i cot 2 "^^ -l)7i/4n) can be represented as integer-valued polynomials in n of degrees 2m -1 and 2m, respectively. Properties of these polynomials are discussed, and recurrence relations for the coefficients are obtained. The proofs of the results depend on the representations of particular polynomials of degree n -1 or less as their own Lagrange interpolation polynomials based on the zeros of the nth Chebyshev polynomial T.(x) = cos(narccosx), -1 < x < 1.

A quantitative version of a classical result of S.N. Bernstein concerning the divergence of Lagrange interpolation polynomials based on equidistant nodes is presented. The proof is motivated by the results of numerical computations.

For the twice-punctured unit disc U p = {z : |z| < 1, 2 ^ ±p} , where 0 < p < 1, we obtain precise descriptions for p near 0 of various parameters associated with the uniformisation of n p by the upper half-plane U = {T : Im T > 0}. These parameters include the hyperbolic length of the geodesic surrounding ±p, the so-called "accessory parameters", and the "proximity parameter" which determines the behaviour of the hyperbolic density near the punctures of fi p .

Given f ∈ C [−1, 1], let Hn, 3(f, x) denote the (0,1,2) Hermite-Fejér interpolation polynomial of f based on the Chebyshev nodes. In this paper we develop a precise estimate for the magnitude of the approximation error |Hn, 3(f, x) − f(x)|. Further, we demonstrate a method of combining the divergent Lagrange and (0,1,2) interpolation methods on the Chebyshev nodes to obtain a convergent rational interpolatory process.