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