On conjectures by Csordas, Charalambides and Waleffe
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- by Alexander Dyachenko and Galina van Bevern PDF
- Proc. Amer. Math. Soc. 144 (2016), 2037-2052 Request permission
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.References
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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
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2037-2052
- MSC (2010): Primary 33C45, 26C10; Secondary 30C15
- DOI: https://doi.org/10.1090/proc/12861
- MathSciNet review: 3460165