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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

Symmetries for Casorati determinants of classical discrete orthogonal polynomials


Author: Antonio J. Durán
Journal: Proc. Amer. Math. Soc. 142 (2014), 915-930
MSC (2010): Primary 42C05, 33C45, 33E30
Published electronically: November 21, 2013
MathSciNet review: 3148526
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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$.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11802-2
Keywords: Orthogonal polynomials, difference equations, discrete classical polynomials, Casorati determinants, Charlier polynomials, Meixner polynomials, Krawtchouk polynomials, Hahn polynomials.
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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.