σ-orthogonal polynomials on the semicircle

Abstract

Orthogonal polynomials on the semicircle and the corresponding quadratures of Gaussian type were introduced and studied by Gautschi and Milovanovi´c for the first time in [1]. In the real case quadrature rules with multiple nodes [3] are observed. A special case of polynomials whose zeros are these multiple nodes are given by the so-called s-orthogonality conditions [2]. The s-orthogonal polynomials on the semicircle are considered in [4]. In this paper we introduce σ-orthogonal polynomials on the semicircle and give a detailed study in three special cases where the weight function is the Chebyshev weight function of the first kind. References [1] W. Gautschi, G. V. Milovanovi´c, Polynomials orthogonal on the semicircle, J. Approx. Theory 46 (1986), pp. 230{250. [2] W. Gautschi, G. V. Milovanovi´c, S-orthogonality and construction of Gauss-Tur´an-type quadrature formulae, J. Comput. Appl. Math. 86, pp. (1997), 205{218. [3] G. V. Milovanovi´c, M. P. Stani´c, Quadrature rules with multiple nodes, In: Mathematical Analysis, Approximation Theory and Their Application, Series: Springer Optimization and its Applications, Vol. 111 (Th. Rassias, V. Gupta, eds.), pp. 435{462, Springer, 2016, ISBN 978-3-319- 31279-8. [4] M. Stani´c, S-orthogonality on the semicircle, Kragujevac J. Math. 26 (2004), pp. 39{48.