On the zeros of Stieltjes and Van Vleck polynomials
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- by Neyamat Zaheer and Mahfooz Alam PDF
- Trans. Amer. Math. Soc. 229 (1977), 279-288 Request permission
Abstract:
Stieltjes and Van Vleck polynomials arise in the study of the polynomial solutions of the generalized Lamé differential equation. Our object is to generalize a theorem due to Marden on the location of the zeros of Stieltjes and Van Vleck polynomials. In fact, our generalization is two-fold: Firstly, we employ sets which are more general than the ones used by Marden for prescribing the location of the complex constants occurring in the Lamé differential equation; secondly, Marden deals only with the standard form of the said differential equation, whereas our result is equally valid for yet another form of the same differential equation. The part of our main theorem concerning Stieltjes polynomials may also be regarded as a generalization of Lucas’ theorem to systems of partial fraction sums.References
- 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 E. Heine, Handbuch der Kugelfuntionen, Bd. I, 2nd ed., Springer, Berlin, 1878, pp. 472-476. F. Lucas, Propriétés géométriques des fractions rationnelle, C. R. Acad. Sci. Paris 77 (1874), 431-433; ibid. 78 (1874), 140-144; ibid. 78 (1874), 180-183; ibid. 78 (1874), 271-274. —, Géométrie des polynômes, J. Ecole Polytech. (1) 46 (1879), 1-33. —, Statique des polynômes, Bull. Soc. Math. France 17 (1888), 2-69.
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 229 (1977), 279-288
- MSC: Primary 30A08
- DOI: https://doi.org/10.1090/S0002-9947-1977-0435367-3
- MathSciNet review: 0435367