×
Sadjang, P. Njionou; Sawalda, D. Kalda; Mboutngam, S.; Foupouagnigni, M.; Koepf, W.

On non-linear characterizations of classical orthogonal polynomials. (English) Zbl 1524.33059

Mediterr. J. Math. 20, No. 1, Paper No. 10, 32 p. (2023).
Summary: Classical orthogonal polynomials are known to satisfy seven equivalent properties, namely the Pearson equation for the linear functional, the second-order differential/difference/\(q\)-differential/ divided-difference equation, the orthogonality of the derivatives, the Rodrigues formula, two types of structure relations, and the Riccati equation for the formal Stieltjes function. In this work, following previous work by K. H. Kwon et al. [J. Difference Equ. Appl. 4, No. 2, 145–162 (1998; Zbl 0933.33011); Kyungpook Math. J. 38, No. 2, 259–281 (1998; Zbl 0922.33005)], we state and prove a non-linear characterization result for classical orthogonal polynomials on non-uniform lattices. Next, we give explicit relations for some families of these classes.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)

Keywords:

classical orthogonal polynomials on non-uniform lattices; difference equation; divided-difference equation; three-term recurrence relation

Citations:

Zbl 0933.33011; Zbl 0922.33005

Software:

OPQ
Cite Review PDF
Full Text: DOI
OA License

References:

[1] Al-Salam, W.A.: Characterization theorems for orthogonal polynomials. In: Nevai, P. (ed.) Orthogonal Polynomials: Theory and Practice, NATO ASI Series C: Math. and Phys. Sciences, vol. 294. Kluwer, Dordrecht, pp. 1-24 (1990) · Zbl 0704.42021
[2] Al-Salam, WA, On the Bessel polynomials, Boll. Un. Mat. Ital. (3), 12, 227-229 (1957) · Zbl 0080.05302
[3] Al-Salam, WA, The Bessel polynomials, Duke Math. J., 24, 529-545 (1957) · Zbl 0085.28901 · doi:10.1215/S0012-7094-57-02460-2
[4] Al-Salam, WA; Chihara, TS, Another characterization of the classical orthogonal polynomials, SIAM J. Math. Anal., 3, 65-70 (1972) · Zbl 0238.33010 · doi:10.1137/0503007
[5] Atakishiyev, NM; Rahman, M.; Suslov, SK, On classical orthogonal polynomials, Constr. Approx., 11, 181-226 (1995) · Zbl 0837.33010 · doi:10.1007/BF01203415
[6] Brezinski, C., Padé-Type Approximation and General Orthogonal Polynomials, International Series of Numerical Mathematics (1980), Basel: Birkhäuser, Basel · Zbl 0418.41012 · doi:10.1007/978-3-0348-6558-6
[7] Chihara, TS, An Introduction to Orthogonal Polynomials (1978), New York: Gordon and Breach, New York · Zbl 0389.33008
[8] Foupouagnigni, M.: On difference and differential equations for modifications of classical orthogonal polynomials. Habilitation Thesis, Fachbereich Mathematik der Universität Kassel (2006)
[9] Foupouagnigni, M., On difference equations for orthogonal polynomials on non-uniform lattices, J. Differ. Equ. Appl., 14, 127-174 (2008) · Zbl 1220.33017 · doi:10.1080/10236190701536199
[10] Foupouagnigni, M., Mboutngam, S.: On the polynomial solution of divided-difference equations of hypergeometric type on nonuniform lattices. Axioms, 8 (2019) · Zbl 1432.33016
[11] Foupouagnigni, M.; Kenfack-Nangho, M.; Mboutngam, S., Characterization theorem of classical orthogonal polynomials on non-uniform lattices: the functional approach, Integr. Transf. Spec. Funct., 22, 10, 739-758 (2011) · Zbl 1234.33012 · doi:10.1080/10652469.2010.546996
[12] Foupouagnigni, M.; Koepf, W.; Kenfack-Nangho, M.; Mboutngam, S., On solutions of holonomic divided-difference equations on non-uniform lattices, Axioms, 3, 404-434 (2013) · Zbl 1301.33024 · doi:10.3390/axioms2030404
[13] García, AG; Marcellán, F.; Salto, L., A distributional study of discrete classical orthogonal polynomials, J. Comput. Appl. Math., 5, 7, 147-162 (1995) · Zbl 0853.33009 · doi:10.1016/0377-0427(93)E0241-D
[14] Gautschi, W., Orthogonal Polynomials: Computation and Approximation (2004), New York: Oxford University Press, New York · Zbl 1130.42300 · doi:10.1093/oso/9780198506720.001.0001
[15] Gautschi, W., Orthogonal polynomials, quadrature, and approximation: computational methods and software (in Matlab), Orthogonal Polynomials and Special Functions. Lecture Notes in Mathematics, 1-77 (2006), Berlin: Springer, Berlin · Zbl 1101.41029 · doi:10.1007/978-3-540-36716-1_1
[16] Kenfack-Nangho, M.: On Laguerre-Hahn orthogonal polynomials on non-uniform lattices. PhD Thesis, Université de Yaoundé I (2014)
[17] Koekoek, R., Swarttouw, R.: The Askey-scheme of hypergeometric orthogonal polynomials and its \(q\)-analogue, Report no. 98-17 (1998). Faculty of Information technology and Systems, Delft University of Technology
[18] Koekoek, R.; Lesky, PA; Swarttouw, RF, Hypergeometric Orthogonal Polynomials and their \(q\)-Analogues. Springer Monographs in Mathematics (2010), Berlin: Springer, Berlin · Zbl 1200.33012 · doi:10.1007/978-3-642-05014-5
[19] Koepf, W.; Schmersau, D., On a structure formula for classical q-orthogonal polynomials, J. Comput. Appl. Math., 136, 99-107 (2001) · Zbl 1004.33008 · doi:10.1016/S0377-0427(00)00577-X
[20] Krall, HL; Frink, O., A new class of orthogonal polynomials: the Bessel polynomials, Trans. Am. Math. Soc., 65, 100-115 (1949) · Zbl 0031.29701 · doi:10.1090/S0002-9947-1949-0028473-1
[21] Kwon, KH; Lee, DW; Park, SB; Yoo, BH, Discrete classical orthogonal polynomials, J. Differ. Equ. Appl., 4, 145-162 (1998) · Zbl 0933.33011 · doi:10.1080/10236199808808134
[22] Kwon, KH; Lee, DW; Park, SB; Yoo, BH, Hahn class orthogonal polynomials, Kyungpook Math. J., 38, 259-281 (1998) · Zbl 0922.33005
[23] Marcellán, F.; Branquinho, AP; Petronilho, J., Classical orthogonal polynomials: a functional approach, Acta Appl. Math., 34, 3, 283-303 (1994) · Zbl 0793.33009 · doi:10.1007/BF00998681
[24] Mboutngam, S.: On semi-classical orthogonal polynomials on non-uniform lattices. PhD Thesis, Université de Yaoundé I (2014)
[25] McCarthy, PJ, Characterizations of classical orthogonal polynomials, Port. Math., 20, 47-52 (1961) · Zbl 0188.12401
[26] Njionou Sadjang, P.: Moments of classical orthogonal polynomials. PhD thesis. Universität Kassel (2013). http://nbn-resolving.de/urn:nbn:de:hebis:34-2013102244291
[27] Njionou Sadjang, P.; Koepf, W.; Foupouagnigni, M., On structure formulas for Wilson polynomials, Integr. Transf. Spec. Funct., 26, 12, 1000-1014 (2015) · Zbl 1331.33011 · doi:10.1080/10652469.2015.1076408
[28] Njionou Sadjang, P.; Koepf, W.; Foupouagnigni, M., On moments of classical orthogonal polynomials, J. Math. Anal. Appl., 424, 122-151 (2015) · Zbl 1321.33022 · doi:10.1016/j.jmaa.2014.10.087
[29] Suslov, SK, The theory of difference analogues of special functions of hypergeometric type, Russ. Math. Surv., 44, 227-278 (1989) · Zbl 0685.33013 · doi:10.1070/RM1989v044n02ABEH002045
[30] Tcheutia, DD; Njionou Sadjang, P.; Koepf, W.; Foupouagnigni, M., Divided-difference equation, inversion, connection, multiplication and linearization formulae of the Continuous Hahn and the Meixner-Pollaczek polynomials, Ramanujan J., 45, 33-56 (2018) · Zbl 1382.33008 · doi:10.1007/s11139-016-9870-5
[31] Trefethen, LN, Approximation Theory and Approximation Practice (2020), Philadelphia: Society for Industrial and Applied Mathematics (SIAM), Philadelphia · Zbl 1437.41001
[32] Tricomi, F.G.: Vorlesungen über Orthogonalreihen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band LXXVI. Springer, Berlin (1955) · Zbl 0065.29601
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.