On the Chebyshev polynomials

We put special attention in this paper on the Chebyshev polynomials of the fourth kind because they are much less known and less studied than others. The representation problem of analytic functions in series of such polynomials is considered, and the important role of the Chebyshev functions of the second kind in solving them is emphasized. For analytic functions, the remainder term of Gauss quadrature rules can be represented as a contour integral with a complex kernel function. The kernel function related to the Gauss quadrature for Chebyshev polynomials of the fourth kind is especially studied on elliptic contours and the points of its maximum are specified.