Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


Casorati type determinants of some $ \mathfrak{q}$-classical orthogonal polynomials

Authors: Antonio J. Durán and Jorge Arvesú
Journal: Proc. Amer. Math. Soc. 144 (2016), 1655-1668
MSC (2010): Primary 42C05, 33C45
Published electronically: September 9, 2015
MathSciNet review: 3451241
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Some symmetries for Casorati determinants whose entries are $ \mathfrak{q}$-classical orthogonal polynomials are studied. Special attention is paid to the symmetry involving Big $ \mathfrak{q}$-Jacobi polynomials. Some limiting situations, for other related $ \mathfrak{q}$-classical orthogonal polynomial families in the $ \mathfrak{q}$-Askey scheme, namely $ \mathfrak{q}$-Meixner, $ \mathfrak{q}$-Charlier, and $ \mathfrak{q}$-Laguerre polynomials, are considered.

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

  • [1] Dimitar K. Dimitrov, Higher order Turán inequalities, Proc. Amer. Math. Soc. 126 (1998), no. 7, 2033-2037. MR 1459117 (99g:33021),
  • [2] Antonio J. Durán, Symmetries for Casorati determinants of classical discrete orthogonal polynomials, Proc. Amer. Math. Soc. 142 (2014), no. 3, 915-930. MR 3148526,
  • [3] Antonio J. Durán, Wronskian type determinants of orthogonal polynomials, Selberg type formulas and constant term identities, J. Combin. Theory Ser. A 124 (2014), 57-96. MR 3176191,
  • [4] Antonio J. Durán, Mario Pérez, and Juan L. Varona, Some conjectures on Wronskian and Casorati determinants of orthogonal polynomials, Exp. Math. 24 (2015), no. 1, 123-132. MR 3305045,
  • [5] G. Felder, A. D. Hemery, and A. P. Veselov, Zeros of Wronskians of Hermite polynomials and Young diagrams, Phys. D 241 (2012), no. 23-24, 2131-2137. MR 2998116,
  • [6] Lun Zhang and Galina Filipuk, On certain Wronskians of multiple orthogonal polynomials, SIGMA Symmetry Integrability Geom. Methods Appl. 10 (2014), Paper 103, 19. MR 3298997,
  • [7] Mourad E. H. Ismail, Determinants with orthogonal polynomial entries, J. Comput. Appl. Math. 178 (2005), no. 1-2, 255-266. MR 2127884 (2006h:33007),
  • [8] Mourad E. H. Ismail and Andrea Laforgia, Monotonicity properties of determinants of special functions, Constr. Approx. 26 (2007), no. 1, 1-9. MR 2310684 (2008e:33001),
  • [9] Samuel Karlin and James McGregor, Coincidence properties of birth and death processes, Pacific J. Math. 9 (1959), 1109-1140. MR 0114247 (22 #5071)
  • [10] S. Karlin and G. Szegö, On certain determinants whose elements are orthogonal polynomials, J. Analyse Math. 8 (1960/1961), 1-157. MR 0142972 (26 #539)
  • [11] Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw, Hypergeometric orthogonal polynomials and their $ q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder. MR 2656096 (2011e:33029)
  • [12] A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical orthogonal polynomials of a discrete variable, Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991. Translated from the Russian. MR 1149380 (92m:33019)
  • [13] Satoru Odake and Ryu Sasaki, Krein-Adler transformations for shape-invariant potentials and pseudo virtual states, J. Phys. A 46 (2013), no. 24, 245201, 24. MR 3064381,
  • [14] Satoru Odake and Ryu Sasaki, Casoratian identities for the Wilson and Askey-Wilson polynomials, J. Approx. Theory 193 (2015), 184-209. MR 3324569,
  • [15] Paul Turán, On the zeros of the polynomials of Legendre, Časopis Pěst. Mat. Fys. 75 (1950), 113-122 (English, with Czech summary). MR 0041284 (12,824g)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 42C05, 33C45

Retrieve articles in all journals with MSC (2010): 42C05, 33C45

Additional Information

Antonio J. Durán
Affiliation: Departamento de Análisis Matemático, Universidad de Sevilla, Apdo (P. O. BOX) 1160, 41080 Sevilla, Spain

Jorge Arvesú
Affiliation: Department of Mathematics, Universidad Carlos III de Madrid, Avda. de la Universidad, 30, 28911, Leganés, Madrid, Spain

Keywords: Orthogonal polynomials, $q$-classical polynomials, Wronskian determinant, Casorati determinant
Received by editor(s): December 4, 2014
Received by editor(s) in revised form: April 28, 2015
Published electronically: September 9, 2015
Additional Notes: This work was partially supported by MTM2012-36732-C03-03 (Ministerio de Economía y Competitividad), FQM-262, FQM-4643, FQM-7276 (Junta de Andalucía) and Feder Funds (European Union). The research of the second author was partially supported by the project MTM2012-36732-C03-01 (Ministerio de Economía y Competitividad)
Communicated by: Walter Van Assche
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society