Dunkl-Appell \(d\)-orthogonal polynomials. (English) Zbl 1137.42005
Here a characterization of the Dunkl-Appell \(d\)-orthogonal polynomials is obtained by means of a generating function. Various properties of this polynomials such as \(a\) \((d+1)\)-order recurrence relation and a differential-difference equation are also obtained. The \(d\)-dimensional functional for which the \(d\)-orthogonality holds is stated. A detailed discussion of the \((d+1)\)-symmetric solutions is presented. Some relationships between the obtained polynomials and some known orthogonal polynomials are also established.
Reviewer: Som Prakash Goyal (Jaipur)
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
Keywords:
Dunkl operator; generating functions; \(d\)-dimensional functional; \(r\)-symmetric sequenceReferences:
| [1] | De Bruin M.G., Polynômes Orthogonaux et Applications, Lecture Notes in Mathematics 1171 pp 74– (1985) |
| [2] | DOI: 10.1016/j.apnum.2004.03.003 · Zbl 1059.41005 · doi:10.1016/j.apnum.2004.03.003 |
| [3] | DOI: 10.1016/S0377-0427(98)00175-7 · Zbl 0958.42015 · doi:10.1016/S0377-0427(98)00175-7 |
| [4] | DOI: 10.1090/S0002-9947-03-03330-0 · Zbl 1033.33002 · doi:10.1090/S0002-9947-03-03330-0 |
| [5] | DOI: 10.1016/S0377-0427(02)00597-6 · Zbl 1021.33006 · doi:10.1016/S0377-0427(02)00597-6 |
| [6] | DOI: 10.1016/S0377-0427(00)00503-3 · Zbl 0969.33005 · doi:10.1016/S0377-0427(00)00503-3 |
| [7] | DOI: 10.1080/10652460008819257 · Zbl 0959.42016 · doi:10.1080/10652460008819257 |
| [8] | Maroni P., Annales dela Faculté des Sciences de Toulouse Mathematices 5 pp 105– (1989) |
| [9] | Van Iseghem J., Journal of Computational and Applied Mathematics 19 pp 141– (1987) |
| [10] | Ben Cheikh Y., Bulletin of the Belgian Mathematical Society 7 pp 107– (2000) |
| [11] | Ben Cheikh Y., Bulletin of the Belgian Mathematical Society. 8 pp 591– (2001) |
| [12] | DOI: 10.1016/S0377-0427(02)00914-7 · Zbl 1055.33005 · doi:10.1016/S0377-0427(02)00914-7 |
| [13] | DOI: 10.1016/j.cam.2005.01.051 · Zbl 1119.42009 · doi:10.1016/j.cam.2005.01.051 |
| [14] | DOI: 10.1215/S0012-7094-68-03551-5 · Zbl 0174.10902 · doi:10.1215/S0012-7094-68-03551-5 |
| [15] | DOI: 10.1016/0377-0427(95)00211-1 · Zbl 0863.33007 · doi:10.1016/0377-0427(95)00211-1 |
| [16] | Ben Cheikh Y., Comptes Rendues del’Académie des Sciences-Series I, Paris 331 pp 349– (2000) · Zbl 1002.33003 · doi:10.1016/S0764-4442(00)01661-X |
| [17] | Zaghouani A., Georgian Mathematical Journal 12 pp 583– (2005) |
| [18] | DOI: 10.1002/mana.19490020103 · Zbl 0031.39001 · doi:10.1002/mana.19490020103 |
| [19] | Rosenblum M., Operator Theory: Advances and Application 73 pp 369– (1994) |
| [20] | Appell P., Annales Scientifiques de ’ École Normale Supériéme 9 pp 119– (1880) |
| [21] | Angelesco A., Buletinul Societâtii Stiite din Cluj 1 pp 44– (1921) |
| [22] | DOI: 10.2307/2370962 · Zbl 0014.30802 · doi:10.2307/2370962 |
| [23] | Al-Salam W. A., Orthogonal Polynomials: Theory and Practice. NATO ASI Series C 294 pp 1– (1990) · Zbl 0704.42021 |
| [24] | DOI: 10.1215/S0012-7094-62-02907-1 · Zbl 0108.06504 · doi:10.1215/S0012-7094-62-02907-1 |
| [25] | Ben Cheikh Y., Applied Mathematics and Computation (2006) |
| [26] | DOI: 10.4153/CJM-1991-069-8 · Zbl 0827.33010 · doi:10.4153/CJM-1991-069-8 |
| [27] | DOI: 10.2307/2154950 · Zbl 0829.33010 · doi:10.2307/2154950 |
| [28] | Freeman J. M., Congressus Numerantium 48 pp 115– (1985) |
| [29] | Szegö, G. 1975.Orthogonal polynomials, 4th, Edited by: Szegö, G. Vol. 23, i–xiii. American Mathematical Society Colloquium Publications. 1–432 · Zbl 0305.42011 |
| [30] | DOI: 10.1016/S0377-0427(01)00423-X · Zbl 0994.33008 · doi:10.1016/S0377-0427(01)00423-X |
| [31] | Ben Cheikh Y., Georgian Mathematical Journal 9 pp 413– (2002) |
| [32] | Chihara, T. S. 1978.An Introduction to Orthogonal Polynomials, i–xii. New York: Gordon and Breach. 1–249 |
| [33] | Ben Cheikh Y., Annales Universitatis Mariae Curie 2 pp 15– (1998) |
| [34] | Ricci P. E., Pubblicazioni dell’Istituto di Matematica Applicata Facolta di Ingegneria Universita degli Studi di 12 pp 27– (1978) |
| [35] | Douak K., Analysis 12 pp 71– (1992) · Zbl 0767.33004 · doi:10.1524/anly.1992.12.12.71 |
| [36] | DOI: 10.1090/S0002-9939-1951-0045875-2 · doi:10.1090/S0002-9939-1951-0045875-2 |
| [37] | Álvarez-Nodarse R., Boletix Sociedad Matemática Mexicana 8 pp 127– (2002) |
| [38] | DOI: 10.1006/jmaa.1997.5771 · Zbl 0893.33009 · doi:10.1006/jmaa.1997.5771 |
| [39] | DOI: 10.1023/A:1014597619994 · Zbl 1003.33008 · doi:10.1023/A:1014597619994 |
| [40] | Koekoek R., Delft University Technology (1998) |
| [41] | DOI: 10.1016/S0377-0427(00)00647-6 · Zbl 1008.42020 · doi:10.1016/S0377-0427(00)00647-6 |
| [42] | Endl K., Annales Scientifiques de ’ École Normale Supérime 3 pp 1– (1956) |
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.
