A new property of a class of Jacobi polynomials

Csordas, George; Charalambides, Marios; Waleffe, Fabian

doi:10.1090/S0002-9939-05-07898-6

A new property of a class of Jacobi polynomials
HTML articles powered by AMS MathViewer

by George Csordas, Marios Charalambides and Fabian Waleffe PDF

Proc. Amer. Math. Soc. 133 (2005), 3551-3560 Request permission

Abstract:

Polynomials whose coefficients are successive derivatives of a class of Jacobi polynomials evaluated at $x=1$ are stable. This yields a novel and short proof of the known result that the Bessel polynomials are stable polynomials. Stability-preserving linear operators are discussed. The paper concludes with three open problems involving the distribution of zeros of polynomials.

References

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.
George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, and Thomas A. Zang, Spectral methods in fluid dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988. MR 917480, DOI 10.1007/978-3-642-84108-8
Thomas Craven and George Csordas, Complex zero decreasing sequences, Methods Appl. Anal. 2 (1995), no. 4, 420–441. MR 1376305, DOI 10.4310/MAA.1995.v2.n4.a4
T. Craven and G. Csordas, Composition theorems, multiplier sequences and complex zero decreasing sequences, Value Distribution Theory and Its Related Topics (G. Barsegian, I. Laine and C. C. Yang, eds.), Kluwer Press (to appear)
Thomas Craven and George Csordas, The Gauss-Lucas theorem and Jensen polynomials, Trans. Amer. Math. Soc. 278 (1983), no. 1, 415–429. MR 697085, DOI 10.1090/S0002-9947-1983-0697085-9
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
Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
Emil Grosswald, Bessel polynomials, Lecture Notes in Mathematics, vol. 698, Springer, Berlin, 1978. MR 520397, DOI 10.1007/BFb0063135
David Gottlieb, The stability of pseudospectral-Chebyshev methods, Math. Comp. 36 (1981), no. 153, 107–118. MR 595045, DOI 10.1090/S0025-5718-1981-0595045-1
David Gottlieb and Liviu Lustman, The spectrum of the Chebyshev collocation operator for the heat equation, SIAM J. Numer. Anal. 20 (1983), no. 5, 909–921. MR 714688, DOI 10.1137/0720063
Jürgen Garloff and David G. Wagner, Hadamard products of stable polynomials are stable, J. Math. Anal. Appl. 202 (1996), no. 3, 797–809. MR 1408355, DOI 10.1006/jmaa.1996.0348
Olga Holtz, Hermite-Biehler, Routh-Hurwitz, and total positivity, Linear Algebra Appl. 372 (2003), 105–110. MR 1999142, DOI 10.1016/S0024-3795(03)00501-9
R. M. Hovstad, A short proof of a continued fraction test for the stability of polynomials, Proc. Amer. Math. Soc. 105 (1989), no. 1, 76–79. MR 973839, DOI 10.1090/S0002-9939-1989-0973839-7
B. Ja. Levin, Distribution of zeros of entire functions, Revised edition, Translations of Mathematical Monographs, vol. 5, American Mathematical Society, Providence, R.I., 1980. Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman. MR 589888
Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1937591
Nikola Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963 (German). MR 0164003
George Pólya, Collected papers, Mathematicians of Our Time, Vol. 8, MIT Press, Cambridge, Mass.-London, 1974. Vol. II: Location of zeros; Edited by R. P. Boas. MR 0505094
Earl D. Rainville, Special functions, The Macmillan Company, New York, 1960. MR 0107725
Theodore J. Rivlin, Chebyshev polynomials, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1990. From approximation theory to algebra and number theory. MR 1060735
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
Fabian Waleffe, Homotopy of exact coherent structures in plane shear flows, Phys. Fluids 15 (2003), no. 6, 1517–1534. MR 1977897, DOI 10.1063/1.1566753