Discrete orthogonal polynomials on Gauss-Lobatto Chebyshev nodes (Q868829)

This paper deals with explicit formulas for discrete orthogonal polynomials over the so-called Gauss-Lobatto Chebyshev nodes \[ X_n=\{x_k=-\cos((k-1)\pi/(n-1))\},\quad k=1, 2, \ldots ,n. \] The orthogonal polynomials \(p_k(x)\) (\(k=1, 2, \dots ,n\)) with respect to the discrete inner product \(\langle f,g\rangle=\sum_{k=1}^nf(x_k)g(x_k)\) on the set \(X_n\) can be related to the Chebyshev polynomials of the first and second kinds. The authors derive explicit expressions of these polynomials and obtain the coefficients in the 3-term recurrence relation that they satisfy. They also obtain formulas for the discrete inner product \(\langle p_k, p_k\rangle\) (\(k=1, 2, \dots , n\)). Numerical examples related to least-squares problems are also given.