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On a conjecture by Karlin and Szegö


Authors: Deok H. Kim and Kil H. Kwon
Journal: Proc. Amer. Math. Soc. 124 (1996), 227-231
MSC (1991): Primary 33C45, 42C05
DOI: https://doi.org/10.1090/S0002-9939-96-03144-9
MathSciNet review: 1301033
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Abstract: In 1961, Karlin and Szegö conjectured : If $\{P_n(x)\}_{n=0}^\infty $ is an orthogonal polynomial system and $\{P_n'(x)\}_{n=1}^\infty $ is a Sturm sequence, then $\{P_n(x)\}_{n=0}^\infty $ is essentially (that is, after a linear change of variable) a classical orthogonal polynomial system of Jacobi, Laguerre, or Hermite. Here, we prove that for any orthogonal polynomial system $\{P_n(x)\}_{n=0}^\infty $, $\{P_n'(x)\}_{n=1}^\infty $ is always a Sturm sequence. Thus, in particular, the above conjecture by Karlin and Szegö is false.


References [Enhancements On Off] (What's this?)

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Additional Information

Kil H. Kwon
Email: khkwon@jacobi.kaist.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-96-03144-9
Received by editor(s): August 12, 1994
Communicated by: Hal L. Smith
Article copyright: © Copyright 1996 American Mathematical Society