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On conjectures by Csordas, Charalambides and Waleffe


Authors: Alexander Dyachenko and Galina van Bevern
Journal: Proc. Amer. Math. Soc. 144 (2016), 2037-2052
MSC (2010): Primary 33C45, 26C10; Secondary 30C15
DOI: https://doi.org/10.1090/proc/12861
Published electronically: September 15, 2015
MathSciNet review: 3460165
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Abstract: In the present note we obtain new results on two conjectures by Csordas et al. regarding the interlacing property of zeros of special polynomials. These polynomials came from the Jacobi tau methods for the Sturm-Liouville eigenvalue problem. Their coefficients are the successive even derivatives of the Jacobi polynomials  $ P_n^{(\alpha ,\beta )}$ evaluated at the point one. The first conjecture states that the polynomials constructed from $ P_n^{(\alpha ,\beta )}$ and $ P_{n-1}^{(\alpha ,\beta )}$ are interlacing when $ -1<\alpha <1$ and $ -1<\beta $. We prove it in a range of parameters wider than that given earlier by Charalambides and Waleffe. We also show that within narrower bounds another conjecture holds. It asserts that the polynomials constructed from $ P_n^{(\alpha ,\beta )}$ and $ P_{n-2}^{(\alpha ,\beta )}$ are also interlacing.


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Additional Information

Alexander Dyachenko
Affiliation: Institut für Mathematik, TU-Berlin, Sekr. MA 4-2, Straße des 17. Juni 136, 10623 Berlin, Germany
Email: dyachenk@math.tu-berlin.de

Galina van Bevern
Affiliation: Institut für Mathematik, TU-Berlin, Sekr. MA 4-2, Straße des 17. Juni 136, 10623 Berlin, Germany
Address at time of publication: Institute of Physics and Technology, Department of Higher Math. and Math. Phys., Tomsk Polytechnic University, Lenin Avenue 2/A, 634000 Tomsk, Russia
Email: gvbevern@yandex.com

DOI: https://doi.org/10.1090/proc/12861
Keywords: Jacobi polynomials, interlacing zeros, tau methods, Hurwitz stability, Hermite-Biehler theorem
Received by editor(s): January 26, 2015
Received by editor(s) in revised form: May 27, 2015
Published electronically: September 15, 2015
Additional Notes: This work was financially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement no. 259173.
Communicated by: Walter Van Assche
Article copyright: © Copyright 2015 American Mathematical Society

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