zbMATH — the first resource for mathematics

Applications of the monotonicity of extremal zeros of orthogonal polynomials in interlacing and optimization problems. (English) Zbl 1210.33015
Summary: We investigate monotonicity properties of extremal zeros of orthogonal polynomials depending on a parameter. Using a functional analysis method we prove the monotonicity of extreme zeros of associated Jacobi, associated Gegenbauer and \(q\)-Meixner-Pollaczek polynomials. We show how these results can be applied to prove interlacing of zeros of orthogonal polynomials with shifted parameters and to determine optimally localized polynomials on the unit ball.

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
PDF BibTeX Cite
Full Text: DOI
[1] Chihara, T.S., An introduction to orthogonal polynomials, (1978), Gordon and Breach, Science Publishers New York · Zbl 0389.33008
[2] Dimitrov, D.K.; Rafaeli, F.R., Monotonicity of zeros of Jacobi polynomials, J. approx. theory, 149, 1, 15-29, (2007) · Zbl 1139.33003
[3] Dimitrov, D.K.; Rafaeli, F.R., Monotonicity of zeros of Laguerre polynomials, J. comput. appl. math., 233, 3, 699-702, (2009) · Zbl 1181.33010
[4] Dimitrov, D.K.; Rodrigues, R.O., On the behaviour of zeros of Jacobi polynomials, J. approx. theory, 116, 2, 224-239, (2002) · Zbl 1006.33003
[5] Driver, K.; Jordaan, K.; Mbuyi, N., Interlacing of the zeros of Jacobi polynomials with different parameters, Numer. algor., 49, 143-152, (2008) · Zbl 1169.30002
[6] Dunkl, C.F.; Xu, Y., Orthogonal polynomials of several variables, (2001), Cambridge University Press Cambridge · Zbl 0964.33001
[7] Elbert, A.; Siafarikas, P.D., Monotonicity properties of the zeros of ultraspherical polynomials, J. approx. theory, 97, 31-39, (1999) · Zbl 0923.33004
[8] Gautschi, W., Orthogonal polynomials: computation and approximation, (2004), Oxford University Press · Zbl 1130.42300
[9] Ismail, M.E.H., The variation of zeros of certain orthogonal polynomials, Adv. appl. math., 8, 111-118, (1987) · Zbl 0628.33001
[10] Ismail, M.E.H., Classical and quantum orthogonal polynomials in one variable, (2005), Cambridge University Press
[11] Ismail, M.E.H.; Letessier, J., Monotonicity of zeros of ultraspherical polynomials, (), 329-330
[12] Ifantis, E.K.; Siafarikas, P.D., Differential inequalities and monotonicity properties of the zeros of associated laquerre and Hermite polynomials, Ann. numer. math., 2, 79-91, (1995) · Zbl 0837.33009
[13] Jordaan, K.; Toókos, F., Mixed recurrence relations and interlacing of the zeros of some q-orthogonal polynomials from different sequences, Acta math. hungar., 128, 1-2, 150-164, (2010) · Zbl 1224.26054
[14] Laforgia, A., A monotonic property for the zeros of ultraspherical polynomials, Proc. amer. math. soc., 83, 757-758, (1981) · Zbl 0471.33006
[15] Muldoon, M., Properties of zeros of orthogonal polynomials and related functions, Journal of computational and applied mathematics, 48, 167-186, (1993) · Zbl 0796.33005
[16] Kokologiannaki, C.G.; Siafarikas, P.D.; Stabolas, I.D., Monotonicity properties and inequalities of the zeros of q-associated polynomials, (), 671-726 · Zbl 1055.33014
[17] Siafarikas, P.D., Inequalities for the zeros of the associated ultraspherical polynomials, Math. inequal. appl., 2, 2, 233-241, (1999) · Zbl 0926.33008
[18] Siafarikas, P.D.; Stabolas, I.D.; Velazquez, L., Differential inequalities of functions involving the lowest zero of some associated orthogonal q-polynomials, Integr. transform. spec. funct., 16, 4, 337-376, (2005) · Zbl 1066.33010
[19] Szegő, G., Orthogonal polynomials, (1975), American Mathematical Society Providence, Rhode Island · JFM 65.0278.03
[20] Wimp, J., Explicit formulas for the associated Jacobi polynomials and some applications, Canad. J. math., 39, 983-1000, (1987) · Zbl 0643.33009
[21] Zarzo, A.; Ronveaux, A.; Godoy, E., Fourth-order differential equation satisfied by the associated of any order of all classical orthogonal polynomials. A study of their distribution of zeros, J. comput. appl. math., 49, 349-359, (1993) · Zbl 0795.33001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.