×

zbMATH — the first resource for mathematics

Limit relations between \(q\)-Krall type orthogonal polynomials. (English) Zbl 1096.33011
In this paper the authors continue the work started by R. Álvarez-Nodarse and J. Petronilho, [“On the Krall-type discrete polynomials”, J. Math. Anal. Appl. 295, No. 1, 55–69 (2004; Zbl 1051.33006)] and study several families of \(q\)- Krall type orthogonal polynomials. In particular, they obtain the limits of the \(q\)-Krall type polynomials in the \(q\)-Hahn tableau. In such a way the limit relations among the Krall-type families are established. In the paper under review the authors start their presentation with some preliminaries and the basic parameters of the families considered in this work. In particular, they include the explicit values for the kernels of the corresponding \(q\)-classical polynomials in terms of the polynomials and their \(q\)-derivatives. Then the \(q\)-Krall type orthogonal polynomials are defined and some algebraic properties are deduced for these new families. Finally, the limits of the modified polynomials of the examples considered in the paper are established.

MSC:
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
33C47 Other special orthogonal polynomials and functions
33E30 Other functions coming from differential, difference and integral equations
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Álvarez-Nodarse, R., Polinomios hipergeométricos y q-polinomios, Monografías del seminario matemático “garcía de galdeano”, vol. 26, (2003), Prensas Universitarias de Zaragoza · Zbl 1033.33011
[2] Álvarez-Nodarse, R.; Arvesú, J.; Marcellán, F., Modifications of quasi-definite linear functionals via addition of delta and derivatives of delta Dirac functions, Indag. math. (N.S.), 15, 1-20, (2004) · Zbl 1089.33005
[3] Álvarez-Nodarse, R.; García, A.G.; Marcellán, F., On the properties for modifications of classical orthogonal polynomials of discrete variables, J. comput. appl. math., 65, 3-18, (1995) · Zbl 0865.42023
[4] Álvarez-Nodarse, R.; Marcellán, F., Difference equation for modifications of meixner polynomials, J. math. anal. appl., 194, 250-258, (1995) · Zbl 0834.39005
[5] Álvarez-Nodarse, R.; Marcellán, F., The modification of classical Hahn polynomials of a discrete variable, Integral transform. spec. funct., 3, 243-262, (1995) · Zbl 0849.33007
[6] Álvarez-Nodarse, R.; Marcellán, F., Limit relations between generalized orthogonal polynomials, Indag. math. (N.S.), 8, 295-316, (1997) · Zbl 0898.33005
[7] Álvarez-Nodarse, R.; Marcellán, F.; Petronilho, J., WKB approximation and krall-type orthogonal polynomials, Acta appl. math., 54, 27-58, (1998) · Zbl 0913.33002
[8] Álvarez-Nodarse, R.; Medem, J.C., The q-classical polynomials and the q-Askey and nikiforov – uvarov tableau, J. comput. appl. math., 135, 197-223, (2001) · Zbl 1024.33013
[9] Álvarez-Nodarse, R.; Petronilho, J., On the krall-type discrete polynomials, J. math. anal. appl., 295, 55-69, (2004) · Zbl 1051.33006
[10] Bavinck, H.; Haeringen, H., Difference equations for generalized meixner polynomials, J. math. anal. appl., 184, 453-463, (1994) · Zbl 0824.33005
[11] Bavinck, H.; Koekoek, K., On a difference equation for generalizations of Charlier polynomials, J. approx. theory, 81, 195-206, (1995) · Zbl 0865.33006
[12] Chihara, T.S., Orthogonal polynomials and measures with end point masses, Rocky mountain J. math., 15, 705-719, (1985) · Zbl 0586.33007
[13] Gasper, G.; Rahman, M., Basic hypergeometric series, (1990), Cambridge Univ. Press Cambridge · Zbl 0695.33001
[14] Godoy, E.; Marcellán, F.; Salto, L.; Zarzo, A., Perturbations of discrete semiclassical functionals by Dirac masses, Integral transform. spec. funct., 5, 19-46, (1997) · Zbl 0877.33002
[15] M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia Math. Appl., vol. 98, Cambridge Univ. Press, in press · Zbl 1082.42016
[16] Koekoek, J.; Koekoek, R., On a differential equation for Koornwinder’s generalized Laguerre polynomials, Proc. amer. math. soc., 112, 1045-1054, (1991) · Zbl 0737.33003
[17] R. Koekoek, R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Reports of the Faculty of Technical Mathematics and Informatics, No. 98-17 Delft University of Technology, Delft, 1998
[18] Koornwinder, T.H., Orthogonal polynomials with weight function \((1 - x)^\alpha(1 + x)^\beta + M \delta(x + 1) + N \delta(x - 1)\), Canad. math. bull., 27, 205-214, (1984) · Zbl 0507.33005
[19] Koornwinder, T.H., Compact quantum groups and q-special functions, (), 46-128 · Zbl 0821.17015
[20] Marcellán, F.; Maroni, P., Sur l’adjonction d’une masse de Dirac à une forme régulière et semi-classique, Ann. mat. pura appl. (4), 162, 1-22, (1992) · Zbl 0771.33008
[21] Medem, J.C.; Álvarez-Nodarse, R.; Marcellán, F., On the q-polynomials: a distributional study, J. comput. appl. math., 135, 157-196, (2001) · Zbl 0991.33007
[22] Nikiforov, A.F.; Suslov, S.K.; Uvarov, V.B., Classical orthogonal polynomials of a discrete variable, Springer ser. comput. phys., (1991), Springer-Verlag Berlin · Zbl 0743.33001
[23] Uvarov, V.B., The connection between systems of polynomials that are orthogonal with respect to different distribution functions, USSR comput. math. math. phys., 9, 25-36, (1969) · Zbl 0231.42013
[24] Vinet, L.; Yermolayeva, O.; Zhedanov, A., A method to study the krall and q-krall polynomials, J. comput. appl. math., 133, 647-656, (2001) · Zbl 0990.33011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.