On Stieltjes and Van Vleck polynomials
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- by Neyamat Zaheer PDF
- Proc. Amer. Math. Soc. 60 (1976), 169-174 Request permission
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).References
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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 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 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 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 10.1090/S0002-9947-1977-0435367-3
Additional Information
- © Copyright 1976 American Mathematical Society
- 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