Zeros of Stieltjes and Van Vleck polynomials and applications. (English) Zbl 0589.30004

The prime interest is in locating the zeros of the system of polynomials that arise in the study of the polynomial solutions of the generalized Lamé’s differential equation (GLDE) \[ \{D^ 2_ z+(\sum^{p}_{j=1}\alpha_ j/(z-a_ j))D_ z+\Phi (z)/\prod^{p}_{j=1}(z-\quad a_ j)\}w(z)=0 \] where \(\Phi\) (z) is a polynomial of degree at most p-2 (p\(\geq 2)\) and \(\alpha_ j\) and \(a_ j\) are complex constants. It is known that there are at most \(C(n+p-2,p- 2)\) Van Vleck polynomials \(V(z):=\Phi (z)\) such that the GLDE has a Stieltjes polynomial solution of degree n. The main theorem develops the notion of reflector curves and sets to locate the zeros of these polynomials relative to a prescribed location of the constants. Several classical theorems follow as special cases. Applications to problems arising in physics, fluid mechanics and the location of the complex zeros of Jacobi polynomials are discussed.
Reviewer: Peter McCoy


30C10 Polynomials and rational functions of one complex variable
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
34A05 Explicit solutions, first integrals of ordinary differential equations
Full Text: DOI


[1] Alam, M., Zeros of Stieltjes and Van Vleck polynomials, Trans. Amer. Math. Soc., 252, 197-204 (1979) · Zbl 0417.30009
[2] Bôcher, M., Über die Reihenentwickelungen der Potentialtheorie, ((1894), Teubner: Teubner Leipzig), 215-218
[3] Heine, E., (Handbuch der Kugelfunctionen, Bd. I (1878), Reimer: Reimer Berlin), 472-476
[4] Kellogg, O. D., Foundations of Potential Theory (1953), Dover: Dover New York · Zbl 0053.07301
[5] Klein, F., Über Lineare Differentialgleichungen der zweiten Ordnung, ((1894)), 211-218, Gottingen · JFM 26.0334.01
[6] Levin, B. Ja, Distribution of Zeros of Entire Functions, (Trans. Math. Monographs, Vol. 5 (1964), Amer. Math. Soc.,: Amer. Math. Soc., Providence, R. I.) · Zbl 0152.06703
[7] Lucas, F., Propriétés géométriques des fractions rationnelles, C. R. Acad. Sci. Paris, 78, 271-274 (1874) · JFM 06.0531.03
[8] Marden, M., Geometry of Polynomials, (Math. Surveys, No. 3 (1966), Amer. Math. Soc.,: Amer. Math. Soc., Providence, R. I.) · JFM 56.0118.08
[9] Marden, M., On Stieltjes polynomials, Trans. Amer. Math. Soc., 33, 934-944 (1931) · JFM 57.0414.01
[10] Milne-Thomson, L. M., Theoretical Hydrodynamics (1968), Macmillan Co.,: Macmillan Co., New York · Zbl 0164.55802
[11] Pólya, G., Sur un théorème de Stieltjes, C. R. Acad. Sci. Paris, 155, 767-769 (1912) · JFM 43.0145.04
[12] Rudin, W., Functional Analysis (1973), Tata McGraw-Hill: Tata McGraw-Hill New Delhi · Zbl 0253.46001
[13] Stieltjes, T. J., Sur certain polynômes qui vérifient une équation différentielle, Acta Math., 6-7, 321-326 (1885) · JFM 17.0310.01
[14] Szegö, G., Orthogonal Polynomials, (Amer. Math. Soc. Colloq. Publ., Vol. 23 (1967), Amer. Math. Soc.,: Amer. Math. Soc., Providence, R. I.) · JFM 65.0278.03
[15] Van Vleck, E. B., On the polynomials of Stieltjes, Bull. Amer. Math. Soc., 4, 426-438 (1898) · JFM 29.0283.02
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.