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

Authors:
Luc Vinet and Alexei Zhedanov

Journal:
Proc. Amer. Math. Soc. **129** (2001), 1317-1327

MSC (2000):
Primary 33D45

DOI:
https://doi.org/10.1090/S0002-9939-01-06047-6

Published electronically:
January 8, 2001

MathSciNet review:
1814158

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Abstract | References | Similar Articles | Additional Information

We consider the polynomials obtained from the little -Jacobi polynomials by inserting a discrete mass at in the orthogonality measure. We show that for , the polynomials are eigensolutions of a linear -difference operator of order with polynomial coefficients. This provides a -analog of results recently obtained for the Krall polynomials.

**1.**T. Chihara,*An Introduction to Orthogonal Polynomials*, Gordon and Breach, NY, 1978. MR**58:1979****2.**G. Gasper and M. Rahman,*Basic hypergeometric series*, Cambridge University Press, Cambridge, 1990. MR**91d:33034****3.**Ya. L. Geronimus,*On the polynomials orthogonal with respect to a given number sequence*Zap. Mat. Otdel. Khar'kov. Univers. i NII Mat. i Mehan.**17**(1940), 3-18 (in Russian).**4.**Ya. L. Geronimus,*On the polynomials orthogonal with respect to a given number sequence and a theorem by W.Hahn*, Izv. Akad. Nauk SSSR**4**(1940), 215-228 (in Russian).**5.**F. Alberto Grünbaum and Luc Haine,*The**-version of a theorem of Bochner*, J. Comput. Appl. Math.**68**(1996), 103-114. MR**97m:33005****6.**J. Koekoek and R. Koekoek,*On a differential equation for Koornwinder's generalized Laguerre polynomials*, Proc. Amer. Math. Soc.**112**(1991), 1045-1054. MR**91j:33008****7.**J. Koekoek and R. Koekoek,*Differential equations for generalized Jacobi polynomials*, J. Comput. Appl. Math., to appear.**8.**J. Koekoek, R. Koekoek, and H. Bavinck,*On differential equations for Sobolev-type Laguerre polynomials*. Trans. Amer. Math. Soc. 350 (1998), no. 1, 347-393. MR**98d:33003****9.**R. Koekoek and R.F. Swarttouw,*The Askey-scheme of hypergeometric orthogonal polynomials and its**-analogue*, Faculty of Technical Mathematics and Informatics, Report 98-17, Delft University of Technology.**10.**T. H. Koornwinder,*Orthogonal polynomials with weight function*Can. Math. Bull.**27**(1984), 205-214. MR**85i:33011****11.**A. Zhedanov,*Rational spectral transformations and orthogonal polynomials*, J. Comput. Appl. Math.**85**(1997), 67-86. MR**98h:42026****12.**A. Zhedanov,*A method of constructing Krall's polynomials*, J. Comput. Appl. Math.**107**(1999), no. 1, 1-20. CMP**99:15**

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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:
https://doi.org/10.1090/S0002-9939-01-06047-6

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