On Stieltjes and Van Vleck polynomials

Zaheer, Neyamat

doi:10.1090/S0002-9939-1976-0419898-2

On Stieltjes and Van Vleck polynomials

Author: Neyamat Zaheer
Journal: Proc. Amer. Math. Soc. 60 (1976), 169-174
MSC: Primary 33A70
DOI: https://doi.org/10.1090/S0002-9939-1976-0419898-2
MathSciNet review: 0419898
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Abstract: Stieltjes and Van Vleck polynomials arise in the study of the polynomial solutions of the generalized Lamé differential equation. The problem of determining the location of the zeros of such polynomials has been studied under quite general conditions by Marden. He has obtained (see Trans. Amer. Math. Soc. 33 (1931), 934-944) varied generalizations of certain results proved earlier by Stieltjes, Van Vleck, Bôcher, Klein, and Pólya. Our object in this paper is to study certain aspects of the corresponding problem in relation to yet another form of the generalized Lamé differential equation. Furthermore, applications of our theorems to the standard form of the generalized Lamé differential equation immediately furnish the corresponding results due to Stieltjes, Van Vleck, and Marden (cf. the paper cited above).

M. Bôcher, Öber die Reihenentwickelungen der Potentialtheorie, Göttingen, 1894, pp. 215-218.

E. Heine, Handbuch der Kugelfunctionen, Bd. I, 2nd ed., Springer, Berlin, 1878, pp. 472-476.

F. Klein, Über lineare Differentialgleichungen der zweiten Ordnung, Göttingen, 1894, pp. 211-218.

F. Lucas, Propriétés géométriques des fractions rationnelles, C. R. Acad. Sci. Paris 77 (1874), 431-433; ibid 78 (1874), 140-144; ibid. 78 (1874), 180-183; ibid. 78 (1874), 271-274.

Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
Morris Marden, On Stieltjes polynomials, Trans. Amer. Math. Soc. 33 (1931), no. 4, 934–944. MR 1501624, DOI https://doi.org/10.1090/S0002-9947-1931-1501624-1

G. Pólya, Sur un théorème de Stieltjes, C. R. Acad. Sci. Paris 155 (1912), 767-769.

T. J. Stieltjes, Sur certains polynômes, Acta Math. 6 (1885), no. 1, 321–326 (French). Qui vérifient une équation différentielle linéaire du second ordre et sur la theorie des fonctions de Lamé. MR 1554669, DOI https://doi.org/10.1007/BF02400421
E. B. van Vleck, On the polynomials of Stieltjes, Bull. Amer. Math. Soc. 4 (1898), no. 9, 426–438. MR 1557633, DOI https://doi.org/10.1090/S0002-9904-1898-00531-1
Neyamat Zaheer and Mahfooz Alam, On the zeros of Stieltjes and Van Vleck polynomials, Trans. Amer. Math. Soc. 229 (1977), 279–288. MR 435367, DOI https://doi.org/10.1090/S0002-9947-1977-0435367-3