Zeros of Jacobi polynomials and associated inequalities

A Dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the Degree of Master of Science. Johannesburg 2015.This Dissertation focuses on the Jacobi polynomial. Specifically, it discusses certain aspects of the zeros of the Jacobi polynomial such as the interlacing property and quasiorthogonality. Also found in the Dissertation is a chapter on the inequalities of the zeros of the Jacobi polynomial, mainly those developed by Walter Gautschi

Inequalities For Zeros Of Jacobi Polynomials Via Sturm’s Theorem: Gautschi’s Conjectures

Let (formula presented) be the zeros of Jacobi polynomials (formula presented) arranged in decreasing order on (formula presented) where (formula presented) and (formula presented) arccos (formula presented) Gautschi, in a series of recent papers, conjectured that the inequalities (formula presented) and (formula presented) hold for all (formula presented) and certain values of the parameters α and β. We establish these conjectures for large domains of the (α, β)-plane by using a Sturmian approach.673549563Ahmed, S., Laforgia, A., Muldoon, M.E., On the spacing of the zeros of some classical orthogonal polynomials (1982) J. London. Math. Soc., 25 (2), pp. 246-252Dimitrov, D.K., Sri Ranga, A., Zeros of a family of hypergeometric para-orthogonal polynomials on the unit circle (2013) Math. Nachr., 286, pp. 1778-1791Driver, K., Jordaan, K., Bounds for extreme zeros of some classical orthogonal polynomials (2012) J. Approx. Theory, 164, pp. 1200-1204Gautschi, W., Leopardi, P., Conjectured inequalities for Jacobi polynomials and their largest zeros (2007) Numer. Algoritm., 45, pp. 217-230Gautschi, W., On a conjectured inequality for the largest zero of Jacobi polynomials (2008) Numer. Algoritm., 49, pp. 195-198Gautschi, W., On conjectured inequalities for zeros of Jacobi polynomials (2009) Numer. Algoritm., 50, pp. 93-96Gautschi, W., New conjectured inequalities for zeros of Jacobi polynomials (2009) Numer. Algoritm., 50, pp. 293-296Gautschi, W., Remark on “New conjectured inequalities for zeros of Jacobi polynomials” by Walter Gautschi. Numer. Algorithm. 50, 293–296 (2009), (2011) Numer. Algoritm., 57, p. 511Hesse, K., Sloan, I.H., Worst-case errors in a Sobolev space setting for cubature over the sphere S2 (2005) Bull. Aust. Math. Soc., 71, pp. 81-105Hesse, K., Sloan, I.H., Cubature over the sphere S2 in Sobolev spaces of arbitrary order (2006) J. Approx. Theory, 141, pp. 118-133Koumandos, S., On a conjectured inequality of Gautschi and Leopardi for Jacobi polynomials (2007) Numer. Algoritm., 44, pp. 249-253Leopardi, P.C., Positive weight quadrature on the sphere and monotonicities of Jacobi polynomials (2007) Numer. Algoritm., 45, pp. 75-87Szegő, G., (1975) Orthogonal Polynomials, , Amer. Math. Soc. Coll. Publ., Providence