×
Tounsi, Mohamed Ihsen; Ben Salah, Imed; Khriji, Lotfi

On the symmetric \(H_q\)-semiclassical polynomial sequences of even class. Some examples from the class two. (English) Zbl 1276.33018

Mediterr. J. Math. 10, No. 3, 1293-1316 (2013).
Summary: We deal with the quadratic decomposition of symmetric \(H_q\)-semiclassical polynomial sequences of even class. Some examples from class two are settled. We give an integral and discrete measure representations for each of the considered symmetric forms.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis

Keywords:

quadratic decomposition; \(H_q\) -semiclassical form; integral representations; \(q\)-difference operator; \(q\)-series representations; \(q\)-difference equation
Cite Review PDF
Full Text: DOI

References:

[1] T. S. Chihara, An introduction to Orthogonal polynomials, Gordon and Breach, New York, 1978. · Zbl 0389.33008
[2] Dini J., Maroni P.: Sur la Multiplication D’une Forme Semi-classique Par un polynme, Publ. Sm. Math. Univ. d’Antananarivo 3, 76–89 (1989)
[3] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990. · Zbl 0695.33001
[4] A. Ghressi and L. Khériji, The symmetrical Hq -semiclassical orthogonal polynomials of class one, SIGMA 5 (2009), Paper 076, 22 pp.. · Zbl 1188.33014
[5] Hahn W.: Über orthogonalpolynome, die q-differenzengleichungen genügen, Math. Nachr. 2, 4–34 (1949) · Zbl 0031.39001 · doi:10.1002/mana.19490020103
[6] Khériji L., Maroni P.: The H q - Classical Orthogonal Polynomials, Acta. Appl. Math. 71((1), 49–115 (2002) · Zbl 1003.33008
[7] Khériji L.: An introduction to the H q -semiclassical orthogonal polynomials, Methods Appl. Anal. 10((3), 387–411 (2003)
[8] Maroni P.: Sur la suite de polynmes orthogonaux associes la forme $${u = \(\backslash\)delta_c + \(\backslash\)lambda(x - c)\^-1}$$ L, Period. Math. Hung. 21((3), 223–248 (1990) · Zbl 0732.42015 · doi:10.1007/BF02651091
[9] Maroni P.: Sur la dcomposition quadratique d’une suite de polynmes orthogonaux I, Rev. Math. Pura ed Appl. 6, 19–53 (1990) · Zbl 0715.33009
[10] P. Maroni, Une thorie algbrique des polynmes orthogonaux. Application aux polynmes orthogonaux semi-classiques, in: C. Brezinski, et al. (Eds.), Orthogonal polynomials and Their Applications, IMACS Ann. Comput. Appl. Math. 9, Baltzer, Basel, (1991), 95–130. · Zbl 0944.33500
[11] Maroni P., Mejri M.: Some semiclassical orthogonal polynomials of class one, Eurasian Math. J. 2((2), 108–128 (2011) · Zbl 1251.33007
[12] P. Maroni and M. Ihsen Tounsi, Quadratic decomposition of symmetric semiclassical polynomial sequences of even class. An example from the case s = 2, J. Difference Equ. Appl., Available online, DOI: 10.1080/10236198.2011.579118 . · Zbl 1254.42034
[13] J. C. Medem Roesicke, Polinomios ortogonales q-semiclásicos, Ph.D. Thesis, Universidad Politcnica de Madrid, Escuela Tcnica Superior de Ingenieros de Caminos, Canales y Puertos, 1996.
[14] Sghaier M., Alaya J.: Semi classical form of class s = 2; the symmetric case, when $${\(\backslash\)Phi(0) = 0}$$ , Methods Appl. Anal. 13((4), 387–409 (2006) · Zbl 1214.42050
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.