Abstract
In Padé approximation the study of the location of the poles of the approximants, i.e. the zeros of the denominators, is of some importance. Loosely speaking, one might say that a (compact) set in the complex plane which is zero-free for a sequence of Padé approximants is the set where this sequence converges to the function it is derived from (under some suitable restrictions of course). As the most widely used sequences (e.g. steplines, diagonals) lead to certain "well-balanced" recurrence relations containing three terms, the study of the behaviour of the zeros of polynomials satisfying 3-term recurrence relations without thinking about a connection with any Padé table whatsoever has received some attention (cf. the famous Parabola theorem due to Saff & Varga [9] and Henrici [5], Leopold [6], Runckel [8]). The aim of this paper is to study the location of the zeros of polynomials generated by certain recurrence relations arising from sequences of approximants in a simultaneous Padé table.
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References
Bruin, M.G. de, Generalized C-fractions and a multidimensional Padé table, Thesis, Amsterdam, 1974.
Bruin, M.G. de, Convergence of generalized C-fractions, J. Approx. Theory 24 (1978), 177–207.
Bruin, M.G. de, Generalized Padé tables and some algorithms therein, Proc. 1st French-Polish meeting on Padé approximation and convergence acceleration techniques, Warsaw 1981 (ed. J. Gilewicz), Centre de Physique Théorique C.N.R.S., Marseille, CPT-'81/PE. 1354, May 1982.
Bruin, M.G. de, Some convergence results in simultaneous rational approximation to the set of hypergeometric functions {1F1(1;ci;z)} ni=1 , in "Padé-Seminar 1983" (Vorlesungsreihe SFB 72; H. Werner, H.J. Bünger eds. Bonn, 1983), 95–117 [to appear in LNM].
Henrici, P., Note on a theorem of Saff and Varga, Padé and Rational Approximation, Theory and Applications, Academic Press, New York, 1977, 157–161.
Leopold, E., Approximants de Padé pour les fonctions de classe S et localisation des zéros de certains polynomes, Thèse de 3ième cycle, Univ. de Provence, January 1982.
Mall, J., Grundlagen für eine Theorie der mehrdimensionalen Padéschen Tafel, Inaugural Dissertation, München, 1934.
Runckel, H.-J., Zero-free parabolic regions for polynomials with complex coefficients, Proc.Am.Math.Soc. 88 (1983), 299–304.
Saff, E.B. and R.S. Varga, Zero-free parabolic regions for sequences of polynomials, SIAM J. Math. Anal. 7 (1976), 344–357.
Saff, E.B. and R.S. Varga, On the zeros and poles of Padé Approximants III, Numer. Math. 30 (1978), 241–266.
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© 1984 Springer-Verlag
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de Bruin, M.G. (1984). Zeros of polynomials generated by 4-term recurrence relations. In: Graves-Morris, P.R., Saff, E.B., Varga, R.S. (eds) Rational Approximation and Interpolation. Lecture Notes in Mathematics, vol 1105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072422
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DOI: https://doi.org/10.1007/BFb0072422
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