Symmetric \(q\)-Dunkl-coherent pairs. (English) Zbl 1537.33013
Summary: In this work, we introduce the notion of a \(q\)-Dunkl-coherent pair of linear functionals in the symmetric case. We prove that if \((u, v)\) is a \(q\)-Dunkl-symmetrically coherent pair of form, then at least one of them must be a \(q\)-Dunkl-classical form. Examples related to the \(q^2\)-analog of generalized Hermite and the \(q^2\)-analog of generalized Gegenbauer forms are given.
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 |
| 44A55 | Discrete operational calculus |
| 39A70 | Difference operators |
| 05A30 | \(q\)-calculus and related topics |
Keywords:
orthogonal polynomials; symmetric polynomials; \(q\)-Dunkl operator; \(q\)-Dunkl-classical orthogonal polynomials; \(q\)-Dunkl-coherent pairReferences:
| [1] | Aloui, B.; Souissi, J., Characterization of q-Dunkl-classical symmetric orthogonal q-polynomials, Ramanujan J., 57, 1355-1365, (2022) · Zbl 1508.33009 · doi:10.1007/s11139-021-00425-8 |
| [2] | Álvarez-Nodarse, R.; Petronilho, J.; Pinzón-Cortés, NC, On linearly related sequences of difference derivatives of discrete orthogonal polynomials, J. Comput. Appl. Math., 284, 26-37, (2015) · Zbl 1312.33021 · doi:10.1016/j.cam.2014.06.018 |
| [3] | Area, I.; Godoy, E.; Marcellán, F., Classification of all \(\Delta \)-coherent pairs, Integral Transforms Spec. Funct., 9, 1-18, (2000) · Zbl 0972.42017 · doi:10.1080/10652460008819238 |
| [4] | Area, I.; Godoy, E.; Marcellán, F., Coherent pairs and orthogonal polynomials of a discrete variable, Integral Transforms Spec. Funct., 14, 31-57, (2003) · Zbl 1047.42019 · doi:10.1080/10652460304546 |
| [5] | Area, I.; Godoy, E.; Marcellán, F., \(q\)-Coherent pairs and \(q\)-orthogonal polynomials, Appl. Math. Comput., 128, 191-216, (2002) · Zbl 1020.33005 |
| [6] | Area, I.: Polinomios ortogonales de variable discreta: pares coherentes. Problemas de conexión [doctoral dissertation]. Universidad de Vigo, España (1999) (in Spanish) |
| [7] | Belmehdi, S., On semi-classical linear functionals of class \(s=1\). Classification and integral representations, Indag. Math. (N.S.), 3, 3, 253-275, (1992) · Zbl 0783.33003 · doi:10.1016/0019-3577(92)90035-J |
| [8] | Bouanani, A.; Khériji, L.; Ihsen Tounsi, M., Characterization of \(q\)-Dunkl Appell symmetric orthogonal \(q\)-polynomials, Expo. Math., 28, 325-336, (2010) · Zbl 1204.33008 · doi:10.1016/j.exmath.2010.03.003 |
| [9] | Bouras, B.; Habbachi, Y.; Marcellán, F., Characterizations of the symmetric \(T_{(\theta , q)}\)-classical orthogonal \(q\)-polynomials, Mediterr. J. Math., 19, 66, (2022) · Zbl 1514.33008 · doi:10.1007/s00009-022-01986-8 |
| [10] | Castillo, K.; Mbouna, D., On another extension of coherent pairs of measures, Indag. Math. (NS)., 31, 2, 223-234, (2020) · Zbl 1434.42038 · doi:10.1016/j.indag.2020.01.001 |
| [11] | Dunkl, CF, Integral kernels with reflection group invariance, Can. J. Math., 43, 1213-1227, (1991) · Zbl 0827.33010 · doi:10.4153/CJM-1991-069-8 |
| [12] | Ghressi, A.; Khériji, L., The symmetrical \(H_q\)-semiclassical orthogonal polynomials of class one, SIGMA 5 J., 11, 076, 22, (2009) · Zbl 1188.33014 |
| [13] | Iserles, A.; Koch, PE; Nørsett, SP, On polynomials orthogonal with respect to certain Sobolev inner products, J. Approx. Theory, 65, 151-175, (1991) · Zbl 0734.42016 · doi:10.1016/0021-9045(91)90100-O |
| [14] | de Jesus, MN; Marcellán, F.; Petronilho, J., \((M, N)\)-coherent pairs of order \((m, k)\) and Sobolev orthogonal polynomials, J. Comput. Appl. Math., 256, 16-35, (2014) · Zbl 1321.33015 · doi:10.1016/j.cam.2013.07.015 |
| [15] | de Jesus, MN; Petronilho, J., On linearly related sequences of derivatives of orthogonal polynomials, J. Math. Anal. Appl., 347, 2, 482-492, (2008) · Zbl 1160.42011 · doi:10.1016/j.jmaa.2008.06.017 |
| [16] | Khériji, L., An introduction to the \(H_q\)-semiclassical orthogonal polynomials, Methods Appl. Anal., 10, 387-411, (2003) · Zbl 1058.33018 · doi:10.4310/MAA.2003.v10.n3.a5 |
| [17] | Khériji, L.; Maroni, P., The \(H_q\)-classical orthogonal polynomials, Acta Appl. Math., 71, 49-115, (2002) · Zbl 1003.33008 · doi:10.1023/A:1014597619994 |
| [18] | Marcellán, F.; Alfaro, M.; Rezola, ML, Orthogonal polynomials on Sobolev spaces: old and new directions, J. Comput. Appl. Math., 48, 113-131, (1993) · Zbl 0790.42015 · doi:10.1016/0377-0427(93)90318-6 |
| [19] | Marcellán, F.; Branquinho, A.; Petronilho, J., Classical orthogonal polynomials: a functional approach, Acta Appl. Math., 34, 283-303, (1994) · Zbl 0793.33009 · doi:10.1007/BF00998681 |
| [20] | Marcellán, F.; Martínez-Finkelshtein, A.; Moreno-Balcazar, JJ, Asymptotics of Sobolev orthogonal polynomials for symmetrically coherent pairs of measures with compact support, J. Comput. Appl. Math., 81, 2, 217-227, (1997) · Zbl 0885.42013 · doi:10.1016/S0377-0427(97)00057-5 |
| [21] | Marcellán, F.; Petronilho, JC, Orthogonal polynomials and coherent pairs: the classical case, Indag. Math. (NS), 6, 287-307, (1995) · Zbl 0843.42010 · doi:10.1016/0019-3577(95)93197-I |
| [22] | Marcellán, F.; Petronilho, JC; Pérez, TE; Piñar, MA, What is beyond coherent pairs of orthogonal polynomials?, J. Comput. Appl. Math., 65, 267-277, (1995) · Zbl 0855.42016 · doi:10.1016/0377-0427(95)00121-2 |
| [23] | Maroni, P., Semi-classical character and finite-type relations between polynomial sequences, Appl. Numer. Math., 31, 295-330, (1999) · Zbl 0962.42017 · doi:10.1016/S0168-9274(98)00137-8 |
| [24] | Maroni, P.: Une théorie algébrique des polynômes orthogonaux: application aux polynômes orthogonaux semi-classiques (in French) [An algebraic theory of orthogonal polynomials: applications to semi-classical orthogonal polynomials]. In: Orthogonal Polynomials and Their Applications (Erice, 1990), pp. 95-130, Annals of Computing and Applied Mathematics, vol. 9, C. Brezinski et al., eds., Baltzer, Basel (1991) · Zbl 0944.33500 |
| [25] | Martínez-Finkelshtein, A.; Moreno Balcázar, JJ; Pérez, TE; Pinãr, MA, Asymptotics of Sobolev orthogonal polynomials for coherent pairs of measures, J. Approx. Theory, 92, 2, 280-293, (1998) · Zbl 0898.42006 · doi:10.1006/jath.1997.3123 |
| [26] | Medem, JC; Álvarez-Nodarse, R.; Marcellán, F., On the \(q\)-polynomials: a distributional study, J. Comput. Appl. Math., 135, 157-196, (2001) · Zbl 0991.33007 · doi:10.1016/S0377-0427(00)00584-7 |
| [27] | Meijer, HG, Determination of all coherent pairs, J. Approx. Theory, 89, 321-343, (1997) · Zbl 0880.42012 · doi:10.1006/jath.1996.3062 |
| [28] | Pan, K., On Sobolev orthogonal polynomials with coherent pairs. The Jacobi case, J. Comput Appl. Math., 79, 2, 249-262, (1997) · Zbl 0871.42024 · doi:10.1016/S0377-0427(96)00168-9 |
| [29] | Pan, K., Asymptotics for Sobolev orthogonal polynomials with coherent pairs: the Jacobi case, type I, Proc. Am. Math. Soc., 126, 2377-2388, (1998) · Zbl 0895.42009 · doi:10.1090/S0002-9939-98-04300-7 |
| [30] | Sghaier, M.; Hamdi, S., Some results on Dunkl-coherent pairs, Integral Transforms Spec. Funct., 32, 11, 863-882, (2021) · Zbl 1500.33005 · doi:10.1080/10652469.2020.1863963 |
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