A discrete approach to monotonicity of zeros of orthogonal polynomials

Abstract

ABSTRACT. We study the monotonicity with respect to a parameter of zeros of orthogonal polynomials. Our method uses the tridiagonal (Jacobi) matrices arising from the three-term recurrence relation for the polynomials. We obtain new results on monotonicity of zeros of associated Laguerre, Al-Salam-Carlitz, Meixner and PoJlaczek polynomials. We also derive inequalities for the zeros of the Al-Salam-Carlitz and Meixner polynomials.

Zeros of classical orthogonal polynomials of a discrete variable

“…Despite the fact that Chebyshev [6,28] emphasized the importance of the zeros of orthogonal polynomials of a discrete variable in 1855, results on monotonicity and limits of their zeros have been obtained only recently in [8,18,24,26,30,31]. In the present paper we establish very sharp estimates for these zeros.…”

“…The behavior of zeros of the classical continuous orthogonal polynomials has been studied extensively, mainly because of their beautiful electrostatic interpretation and their important role as nodes of Gaussian quadrature formulae [2,3,10,12,13,14,18,19,25].…”

Inequalities for Extreme Zeros of Some Classical Orthogonal and q-orthogonal Polynomials

“…Following Ismail's idea, the monotonicity of the extremal zeros of ultraspherical polynomials was investigated by Elbert and Siafarikas [15] while Erb and Tookos [16] applied a version of Ismail's theorem to prove corresponding monotonicity results for zeros of associated Jacobi, associated Gegenbauer and q-Meixner-Pollaczek polynomials. Ismail and Zhang [29] applied the Hellmann-Feynman Theorem while Ismail and Muldoon [30] used the tridiagonal matrices arising from the three term recurrence relation to study monotonicity properties of various special functions and orthogonal polynomials. Dimitrov and Rodrigues [11] invoked the classical Routh-Hurwitz stability criterion to obtain monotonicity results for the zeros of Jacobi polynomials.…”

Monotonicity of zeros of orthogonal Laurent polynomials

“…Theorem 2.3 in [8] provides a powerful tool for testing the monotonic behaviour of zeros of orthogonal polynomials. Since it is not easy to check if a matrix of a general type is positive definite, Ismail and Muldoon used some criteria for positive definiteness of a tridiagonal matrix in order to reformulate their Theorem 2.3 into Theorem 3.3 which gives sufficient conditions for monotonicity of zeros of orthogonal polynomials in terms of chain sequences.…”