A discrete approach to monotonicity of zeros of orthogonal polynomials
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- by Mourad E. H. Ismail and Martin E. Muldoon
- Trans. Amer. Math. Soc. 323 (1991), 65-78
- DOI: https://doi.org/10.1090/S0002-9947-1991-1014251-8
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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 Pollaczek polynomials. We also derive inequalities for the zeros of the Al-Salam-Carlitz and Meixner polynomials.References
- R. Askey, private communication.
- Richard Askey and Mourad Ismail, Recurrence relations, continued fractions, and orthogonal polynomials, Mem. Amer. Math. Soc. 49 (1984), no. 300, iv+108. MR 743545, DOI 10.1090/memo/0300
- Richard Askey and Jet Wimp, Associated Laguerre and Hermite polynomials, Proc. Roy. Soc. Edinburgh Sect. A 96 (1984), no. 1-2, 15–37. MR 741641, DOI 10.1017/S0308210500020412
- F. V. Atkinson, Discrete and continuous boundary problems, Mathematics in Science and Engineering, Vol. 8, Academic Press, New York-London, 1964. MR 0176141
- T. S. Chihara, Chain sequences and orthogonal polynomials, Trans. Amer. Math. Soc. 104 (1962), 1–16. MR 138933, DOI 10.1090/S0002-9947-1962-0138933-7
- T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884 A. Erdélyi et al., Higher transcendental functions, vol. 1, McGraw-Hill, New York, 1953. —, Higher transcendental functions, vol. 2, McGraw-Hill, New York, 1954. G. Freud, Orthogonal polynomials, English transl., Pergamon Press, New York, 1971.
- Wolfgang Hahn, Über Orthogonalpolynome mit drei Parametern, Deutsche Math. 5 (1940), 273–278 (German). MR 13467
- Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183, DOI 10.1017/CBO9780511810817
- Mourad E. H. Ismail, The variation of zeros of certain orthogonal polynomials, Adv. in Appl. Math. 8 (1987), no. 1, 111–118. MR 876957, DOI 10.1016/0196-8858(87)90009-1
- Mourad E. H. Ismail, Monotonicity of zeros of orthogonal polynomials, $q$-series and partitions (Minneapolis, MN, 1988) IMA Vol. Math. Appl., vol. 18, Springer, New York, 1989, pp. 177–190. MR 1019851, DOI 10.1007/978-1-4684-0637-5_{1}4
- Mourad E. H. Ismail, David R. Masson, Jean Letessier, and Galliano Valent, Birth and death processes and orthogonal polynomials, Orthogonal polynomials (Columbus, OH, 1989) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 294, Kluwer Acad. Publ., Dordrecht, 1990, pp. 229–255. MR 1100296, DOI 10.1007/978-94-009-0501-6_{1}1
- Mourad E. H. Ismail, Jean Letessier, and Galliano Valent, Linear birth and death models and associated Laguerre and Meixner polynomials, J. Approx. Theory 55 (1988), no. 3, 337–348. MR 968940, DOI 10.1016/0021-9045(88)90100-1 M. E. H. Ismail, D. Stanton, and D. White, The combinatorics of Charlier polynomials (to appear).
- Mourad E. H. Ismail and Ruiming Zhang, On the Hellmann-Feynman theorem and the variation of zeros of certain special functions, Adv. in Appl. Math. 9 (1988), no. 4, 439–446. MR 968677, DOI 10.1016/0196-8858(88)90022-X
- Andrea Laforgia and Martin E. Muldoon, Some consequences of the Sturm comparison theorem, Amer. Math. Monthly 93 (1986), no. 2, 89–94. MR 827581, DOI 10.2307/2322698
- Lee Lorch, Elementary comparison techniques for certain classes of Sturm-Liouville equations, Differential equations (Proc. Internat. Conf., Uppsala, 1977) Sympos. Univ. Upsaliensis Ann. Quingentesimum Celebrantis, No. 7, Almqvist & Wiksell, Stockholm, 1977, pp. 125–133. MR 0492536
- Albert W. Marshall and Ingram Olkin, Inequalities: theory of majorization and its applications, Mathematics in Science and Engineering, vol. 143, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 552278 G. Szegà, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, 4th ed., Amer. Math. Soc.,, Providence, R.I., 1975.
- H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Co., Inc., New York, N. Y., 1948. MR 0025596
- H. S. Wall and Marion Wetzel, Quadratic forms and convergence regions for continued fractions, Duke Math. J. 11 (1944), 89–102. MR 11340
Bibliographic Information
- © Copyright 1991 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 323 (1991), 65-78
- MSC: Primary 33C50; Secondary 15A42
- DOI: https://doi.org/10.1090/S0002-9947-1991-1014251-8
- MathSciNet review: 1014251