×
Álvarez-Nodarse, R.; Medem, J. C.

\(q\)-classical polynomials and the \(q\)-Askey and Nikiforov-Uvarov tableaus. (English) Zbl 1024.33013

J. Comput. Appl. Math. 135, No. 2, 197-223 (2001).
The authors continue the study of \(q\)-classical orthogonal polynomials started in [J. C. Medem, R. Álvarez-Nodarse and F. Marcellán, J. Comput. Appl. Math. 135, 157-196 (2001; Zbl 0991.33007)]. By using a characterization based on Hahn’s \(q\)-derivative operator a partly classification is obtained for the classical \(q\)-orthogonal polynomials belonging to the \(q\)-Askey scheme of basic hypergeometric orthogonal polynomials. The authors concentrate on the discrete part of this scheme. The results are also compared with those of A. F. Nikiforov and V. B. Uvarov [Int. Trans. Spec. Funct. 1, 223-249 (1993; Zbl 1023.33002)].
Reviewer: Roelof Koekoek (Delft)

Cited in 1 Review
Cited in 20 Documents

MSC:

33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)

Keywords:

\(q\)-orthogonal polynomials; exponential lattices; \(q\)-difference equations

Citations:

Zbl 0991.33007; Zbl 1023.33002
Cite Review PDF
Full Text: DOI

References:

[1] Ahlfors, L., Complex Analysis (1953), McGraw-Hill: McGraw-Hill New York
[2] Álvarez-Nodarse, R.; Arvesú, J., On the \(q\)-polynomials in the exponential lattice \(x(s)=c_1q^s+c_3\), Integral Transform. Spec. Funct., 8, 299-324 (1999) · Zbl 0956.33009
[3] Andrews, G. E., \(q\)-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. \(q\)-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, Conference Series in Mathematics, Vol. 66 (1986), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0594.33001
[4] Andrews, G. E.; Askey, R., Classical orthogonal polynomials, (Brezinski, C.; etal., Polynômes Orthogonaux et Applications. Polynômes Orthogonaux et Applications, Lecture Notes in Mathematics, Vol. 1171 (1985), Springer: Springer Berlı́n), 36-62 · Zbl 0596.33016
[5] Atakishiyev, N. M.; Rahman, M.; Suslov, S. K., On classical orthogonal polynomials, Constr. Approx., 11, 181-226 (1995) · Zbl 0837.33010
[6] Atakishiyev, N. M.; Suslov, S. K., Difference hypergeometric functions, (Gonchar, A. A.; Saff, E. B., Progress in Approximation Theory: An International Perspective. Progress in Approximation Theory: An International Perspective, Springer Series in Computational Mathematics, Vol. 19 (1992), Springer: Springer New York), 1-35 · Zbl 0787.33003
[7] Chihara, T. S., An Introduction to Orthogonal Polynomials (1978), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers New York · Zbl 0389.33008
[8] Dehesa, J. S.; Nikiforov, A. F., The orthogonality properties of \(q\)-polynomials, Integral Transform. Spec. Funct., 4, 343-354 (1996) · Zbl 0867.33014
[9] Fine, N. J., Basic Hypergeometric Series and Applications. Basic Hypergeometric Series and Applications, Mathematical Surveys and Monographs, Vol. 27 (1988), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0647.05004
[10] Gasper, G.; Rahman, M., Basic Hypergeometric Series (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0695.33001
[11] Hahn, W., Über Orthogonalpolynome die \(q\)-differenzengleichungen genügen, Math. Nachr., 2, 4-34 (1949) · Zbl 0031.39001
[12] S. Häcker, Polynomiale Eigenwertprobleme zweiter Ordnung mit Hahnschen \(q\)-Operatoren, Doctoral Dissertation, Universität Stuttgart, Stuttgart, 1993 (in German).; S. Häcker, Polynomiale Eigenwertprobleme zweiter Ordnung mit Hahnschen \(q\)-Operatoren, Doctoral Dissertation, Universität Stuttgart, Stuttgart, 1993 (in German). · Zbl 0859.33018
[13] 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.; 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.
[14] Koornwinder, T. H., Orthogonal polynomials in connection with quantum groups, (Nevai, P., Orthogonal Polynomials, Theory and Practice, Vol. 294 (1990), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 257-292 · Zbl 0697.42019
[15] Koornwinder, T. H., Compact quantum groups and \(q\)-special functions, (Baldoni, V.; Picardello, M. A., Representations of Lie Groups and Quantum Groups. Representations of Lie Groups and Quantum Groups, Pitman Research Notes in Mathematics Series, Vol. 311 (1994), Longman Scientific & Technical: Longman Scientific & Technical New York), 46-128 · Zbl 0821.17015
[16] Maroni, P., Une théorie algébrique des polynômes orthogonaux semiclassiques, (Brezinski, C.; etal., Orthogonal Polynomials and their Applications, IMACS Ann. Comput. Appl. Math. Vol. 9 (1991), Baltzer: Baltzer Basel), 95-130 · Zbl 0944.33500
[17] J.C. Medem, Polinomios \(q\)-semiclásicos, Doctoral Dissertation, Universidad Politécnica. Madrid, 1996 (in Spanish).; J.C. Medem, Polinomios \(q\)-semiclásicos, Doctoral Dissertation, Universidad Politécnica. Madrid, 1996 (in Spanish).
[18] J.C. Medem, R. Álvarez-Nodarse, F. Marcellán, On the \(q\)-polynomials: a distributional study, this issue, J. Comput. Appl. Math. 135 (2001) 157-196.; J.C. Medem, R. Álvarez-Nodarse, F. Marcellán, On the \(q\)-polynomials: a distributional study, this issue, J. Comput. Appl. Math. 135 (2001) 157-196. · Zbl 0991.33007
[19] Medem, J. C.; Marcellán, F., \(q\)-classical polynomials: a very classical approach, Electron. Trans. Numer. Anal., 9, 112-127 (1999) · Zbl 0965.33009
[20] Nikiforov, A. F.; Suslov, S. K.; Uvarov, V. B., Classical Orthogonal Polynomials of a Discrete Variable, Springer Series in Computational Physics (1991), Springer: Springer Berlin · Zbl 0743.33001
[21] A.F. Nikiforov, V.B. Uvarov, Classical orthogonal polynomials in a discrete variable on nonuniform lattices, Preprint Inst. Prikl. Mat. Im. M.V. Keldysha Akad. Nauk SSSR, Moscú, Vol. 17, 1983 (in Russian).; A.F. Nikiforov, V.B. Uvarov, Classical orthogonal polynomials in a discrete variable on nonuniform lattices, Preprint Inst. Prikl. Mat. Im. M.V. Keldysha Akad. Nauk SSSR, Moscú, Vol. 17, 1983 (in Russian).
[22] Nikiforov, A. F.; Uvarov, V. B., Special Functions of Mathematical Physics (1988), Birkhäuser: Birkhäuser Basel · Zbl 0694.33005
[23] Nikiforov, A. F.; Uvarov, V. B., Polynomial solutions of hypergeometric type difference equations and their classification, Integral Transform. Spec. Funct., 1, 223-249 (1993) · Zbl 1023.33002
[24] S.K. Suslov, The theory of difference analogues of special functions of hypergeometric type, Uspekhi Mat. Nauk. 44(2) (1989) 185-226 (English translation Russian Math. Surveys 44(2) (1989) 227-278).; S.K. Suslov, The theory of difference analogues of special functions of hypergeometric type, Uspekhi Mat. Nauk. 44(2) (1989) 185-226 (English translation Russian Math. Surveys 44(2) (1989) 227-278). · Zbl 0681.33019
[25] Vilenkin, N. Ja.; Klimyk, A. U., Representations of Lie Groups and Special Functions, Vol. I,II,III (1992), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0826.22001
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.