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On solutions of linear differential equations with real zeros; proof of a conjecture of Hellerstein and Rossi


Author: Franz Brüggemann
Journal: Proc. Amer. Math. Soc. 113 (1991), 371-379
MSC: Primary 34A20; Secondary 30D35, 34C10
DOI: https://doi.org/10.1090/S0002-9939-1991-1057941-9
MathSciNet review: 1057941
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Abstract: We prove the following conjecture that is due to Hellerstein and Rossi: Let $ \left\{ {{w_1}, \ldots ,{w_n}} \right\}$ be a fundamental system of

$\displaystyle Lw = {w^{(n)}} + {a_{n - 1}}(z){w^{(n - 1)}} + \cdots + {a_0}(z)w \equiv 0$

with polynomials $ {a_j}(z)(0 \leq j \leq n - 1)$. If each $ {w_k}(1 \leq k \leq n)$ has only finitely many nonreal zeros, then there exists a polynomial $ q(z)$ such that $ {u_k}: = \exp (q(z)){w_k}(1 \leq k \leq n)$ form a fundamental system of a homogeneous linear differential equation with constant coefficients.

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  • [1] G. D. Birkhoff, Singular points of ordinary linear differential equations, Trans. Amer. Math. Soc. 10 (1909), 436-470. MR 1500848
  • [2] D. A. Brannan and W. K. Hayman, Research problems in complex analysis, Bull. London Math. Soc. 21 (1989), 1-35. MR 967787 (89m:30001)
  • [3] F. Brüggemann, Untersuchungen linearer Differentialgleichungen mit rationalen Koeffizienten im Komplexen, Dissertation, RWTH, Aachen, 1989.
  • [4] -, On the zeros of fundamental systems of linear differential equations with polynomial coefficients, Complex Variables Theory Appl. (to appear). MR 1074058 (91j:34009)
  • [5] V. Dietrich, Newton-Puiseux diagramm für Systeme linearer Differentialgleichungen, Complex Variables Theory Appl. 7 (1987), 256-296. MR 889115 (88g:34003)
  • [6] -, Über die Annahme der möglichen Wachstumsordnungen und Typen bei linearen Differentialgleichungen, Habilitationsschrift, RWTH, Aachen, 1989.
  • [7] G. Frank, Picardsche Ausnahmewerte bei Lösungen linearer Differentialgleichungen, Manuscripta Math. 2 (1970), 181-190. MR 0264160 (41:8756)
  • [8] G. Gundersen, On the real zeros of solutions to $ f'' + Af = 0$ where $ f$ is entire, Ann. Acad. Sci. Fenn. Ser. A I Math. 11 (1986), 275-294. MR 853961 (87j:34013)
  • [9] S. Hellerstein and J. Rossi, Zeros of meromorphic solutions of second order linear differential equations, Math. Z. 192 (1986), 603-612. MR 847009 (87m:34006)
  • [10] S. Hellerstein, L. C. Shen, and J. Williamson, Real zeros of derivatives of meromorphic functions and solutions of second-order differential equations, Trans. Amer. Math. Soc. 285 (1984), 759-776. MR 752502 (85j:30065)
  • [11] G. Jank and L. Volkmann, Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, Birkhäuser, Basel, Boston, and Stuttgart, 1985. MR 820202 (87h:30066)
  • [12] H. Poincaré, Sur les intégrales des équations linéaires, Acta Math. 8 (1886), 295-344.
  • [13] L. W. Thomé, Zur Theorie der linearen Differentialgleichungen, J. Reine Angew. Math. 75 (1873), 265-291.
  • [14] H. L. Turrittin, Asymptotic distribution of zeros for certain exponential sums, Amer. J. Math. 66 (1944), 199-228. MR 0010216 (5:263c)
  • [15] H. Wittich, Zur Kennzeichnung linearer Differentialgleichungen mit konstanten Koeffizienten, Festband zum 70. Geburtstag von Rolf Nevanlinna (H. P. Künzi and A. Pfluger, eds.), Springer-Verlag, 1966. MR 0229886 (37:5452)

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DOI: https://doi.org/10.1090/S0002-9939-1991-1057941-9
Article copyright: © Copyright 1991 American Mathematical Society

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