Symmetries for Casorati determinants of classical discrete orthogonal polynomials
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- by Antonio J. Durán PDF
- Proc. Amer. Math. Soc. 142 (2014), 915-930 Request permission
Abstract:
Given a classical discrete family $(p_n)_n$ of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn) and the set of numbers $m+i-1$, $i=1,\cdots ,k$ and $k,m\ge 0$, we consider the $k\times k$ Casorati determinants $\det ((p_{n+j-1}(m+i-1))_{i,j=1}^k)$, $n\ge 0$. In this paper, we conjecture a nice symmetry for these Casorati determinants and prove it for the cases $k\ge 0, m=0,1$ and $m\ge 0, k=0,1$. This symmetry is related to the existence of higher order difference equations for the orthogonal polynomials with respect to certain Christoffel transforms of the classical discrete measures. Other symmetry will be conjectured for the Casorati determinants associated to the Meixner and Hahn families and the set of numbers $-c+i$, $i=1,\cdots ,k$ and $k,m\ge 0$.References
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Additional Information
- Antonio J. Durán
- Affiliation: Departamento de Análisis Matemático, Universidad de Sevilla, Apdo (P.O. Box) 1160, 41080 Sevilla, Spain
- Email: duran@us.es
- Received by editor(s): February 15, 2012
- Received by editor(s) in revised form: April 3, 2012, and April 5, 2012
- Published electronically: November 21, 2013
- Additional Notes: The author was partially supported by MTM2009-12740-C03-02 (Ministerio de Economía y Competitividad), FQM-262, FQM-4643, FQM-7276 (Junta de Andalucía), and Feder Funds (European Union).
- Communicated by: Walter Van Assche
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 915-930
- MSC (2010): Primary 42C05, 33C45, 33E30
- DOI: https://doi.org/10.1090/S0002-9939-2013-11802-2
- MathSciNet review: 3148526