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

 

Generalized little $q$-Jacobi polynomials as eigensolutions of higher-order $q$-difference operators


Authors: Luc Vinet and Alexei Zhedanov
Journal: Proc. Amer. Math. Soc. 129 (2001), 1317-1327
MSC (2000): Primary 33D45
Published electronically: January 8, 2001
MathSciNet review: 1814158
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Abstract:

We consider the polynomials $p_n(x;a,b;M)$ obtained from the little $q$-Jacobi polynomials $p_n(x;a, b)$ by inserting a discrete mass $M$ at $x=0$ in the orthogonality measure. We show that for $a=q^j, \; j=0,1,2,\dots$, the polynomials $p_n(x;a,b;M)$ are eigensolutions of a linear $q$-difference operator of order $2j+4$ with polynomial coefficients. This provides a $q$-analog of results recently obtained for the Krall polynomials.


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Additional Information

Luc Vinet
Affiliation: Department of Mathematics and Statistics and Department of Physics, McGill University, 845 Sherbrooke St. W., Montreal, Québec, Canada H3A 2T5 – Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128, succursale Centre-ville, Montréal, Québec, Canada H3C 3J7
Email: vinet@crm.umontreal.ca

Alexei Zhedanov
Affiliation: Donetsk Institute for Physics and Technology, Donetsk 340114, Ukraine
Email: zhedanov@kinetic.ac.donetsk.ua

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06047-6
PII: S 0002-9939(01)06047-6
Keywords: Krall's polynomials, little $q$-Jacobi polynomials
Received by editor(s): December 11, 1998
Published electronically: January 8, 2001
Additional Notes: The work of the first author was supported in part through funds provided by NSERC (Canada) and FCAR (Quebec). The work of the second author was supported in part through funds provided by SCST (Ukraine) Project #2.4/197, INTAS-96-0700 grant and project 96-01-00281 supported by RFBR (Russia). The second author thanks Centre de recherches mathématiques of the Université de Montréal for hospitality.
Communicated by: Hal L. Smith
Article copyright: © Copyright 2001 American Mathematical Society