×

On semi-classical \(d\)-orthogonal polynomials. (English) Zbl 1432.33010

Summary: In this paper a general theory of semi-classical \(d\)-orthogonal polynomials is developed. We define the semi-classical linear functionals by means of a distributional equation \((\Phi\mathcal{U})^\prime=\Psi\mathcal{U}\), where \(\Phi \) and \(\Psi \) are \(d\times d\) matrix polynomials. Several characterizations for these semi-classical functionals are given in terms of the corresponding \(d\)-orthogonal polynomials sequence. They involve a quasi-orthogonality property for their derivatives and some finite-type relations.

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

Software:

na15
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ben Cheikh, On some (n-1)-symmetric linear functionals, J. Comput. Appl. Math. 133 pp 207– (2001) · Zbl 1008.42020 · doi:10.1016/S0377-0427(00)00647-6
[2] Ben Cheikh, Some discrete d-orthogonal polynomial sets, J. Comput. Appl. Math. 156 pp 2– (2003) · Zbl 1055.33005 · doi:10.1016/S0377-0427(02)00914-7
[3] Ben Cheikh, d-orthogonality via generating functions, J. Comput. Appl. Math. 199 pp 253– (2007) · Zbl 1119.42009 · doi:10.1016/j.cam.2005.01.051
[4] Y. Ben Cheikh N. Ben Romdhane d -orthogonal polynomial sets of Chebyshev type S. Elaydi 2005 100 111
[5] Ben Cheikh, On d-symmetric classical d-orthogonal polynomials, J. Comput. Appl. Math. 236 pp 85– (2011) · Zbl 1261.42040 · doi:10.1016/j.cam.2011.03.027
[6] Ben Cheikh, On Askey-scheme and d-orthogonality, I: A characterization theorem, J. Comput. Appl. Math. 233 pp 621– (2009) · Zbl 1188.33018 · doi:10.1016/j.cam.2009.02.029
[7] Ben Cheikh, d-orthogonality of Little q-Laguerre type polynomials, J. Comput. Appl. Math. 236 pp 74– (2011) · Zbl 1232.33028 · doi:10.1016/j.cam.2011.03.006
[8] Ben Cheikh, Some generalized hypergeometric d-orthogonal polynomials, J. Math. Anal. Appl. 343 pp 464– (2008) · Zbl 1140.33003 · doi:10.1016/j.jmaa.2008.01.055
[9] Ben Cheikh, On the classical d-orthogonal polynomials defined by certain generating functions, I, Bull. Belg. Math. Soc. 7 pp 107– (2000) · Zbl 0945.33007
[10] Ben Cheikh, On the classical d-orthogonal polynomials defined by certain generating functions, II, Bull. Belg. Math. Soc. 8 pp 591– (2001) · Zbl 1036.33006
[11] Ben Cheikh, Dunkl-Appell d-orthogonal polynomials, Integral Transforms Spec. Funct. 18 pp 581– (2007) · Zbl 1137.42005 · doi:10.1080/10652460701445302
[12] Ben Sahal, The connection between self-associated two-dimensional vector functionals and third degree forms, Adv. Comput. Math. 13 pp 51– (2000) · Zbl 0943.42013 · doi:10.1023/A:1018941924408
[13] Blel, On m-symmetric d-orthogonal polynomials, C. R. Acad. Sci. Paris, Ser. I 350 pp 19– (2012) · Zbl 1248.33013 · doi:10.1016/j.crma.2011.12.011
[14] Boukhemis, Une caracterisation des polynômes strictement 1/p orthogonaux de type Scheffer. Etude du cas p=2, J. Approx. Theory 54 pp 67– (1988) · Zbl 0662.42018 · doi:10.1016/0021-9045(88)90117-7
[15] Boukhemis, A study of a sequence of classical orthogonal polynomials of dimension 2, J. Approx. Theory 90 pp 435– (1997) · Zbl 0885.42015 · doi:10.1006/jath.1996.3078
[16] Boukhemis, On the classical 2-orthogonal polynomials sequences of sheffer-Meixner type, Cubo. A Math. J. 7 pp 39– (2005) · Zbl 1101.33005
[17] Boukhemis, Classical 2-orthogonal polynomials and differential equations, Int. J. Math. Math. Sci. pp 1– (2006) · Zbl 1116.33007
[18] Chihara, An Introduction to Orthogonal Polynomials (1978) · Zbl 0389.33008
[19] da Rocha, Shohat-Favard and Chebyshev’s methods in d-orthogonality, Numer. Algorithms 20 pp 139– (1999) · Zbl 0941.42011 · doi:10.1023/A:1019151817161
[20] Douak, The relation of the d-orthogonal polynomials to the Appell polynomials, J. Comput. Appl. Math. 70 pp 279– (1996) · Zbl 0863.33007 · doi:10.1016/0377-0427(95)00211-1
[21] Douak, On 2-orthogonal polynomials of Laguerre type, Int. J. Math. Math. Sci. 22 pp 29– (1999) · Zbl 0927.33006 · doi:10.1155/S0161171299220297
[22] Douak, Les polynômes orthogonaux classiques ”de dimension deux, Analysis 12 pp 71– (1992) · Zbl 0767.33004 · doi:10.1524/anly.1992.12.12.71
[23] Douak, Une caractérisation des polyn ômes d-orthogonaux classiques, J. Approx. Theory 82 pp 177– (1995) · Zbl 0849.33004 · doi:10.1006/jath.1995.1074
[24] Douak, On d-orthogonal Tchebychev polynomials, I., Appl. Numer. Math. 24 pp 23– (1997) · Zbl 0881.33024 · doi:10.1016/S0168-9274(97)00006-8
[25] Douak, On d-orthogonal Tchebychev polynomials II, Methods Appl. Anal. 4 (4) pp 404– (1997) · Zbl 0904.33003
[26] Drake, Higher-order matching polynomials and d-orthogonality, Adv. Appl. Math. 46 pp 226– (2011) · Zbl 1227.05256 · doi:10.1016/j.aam.2009.12.008
[27] Genest, d-orthogonal polynomials and su(2), J. Math. Anal. Appl. 390 pp 472– (2012) · Zbl 1238.33004 · doi:10.1016/j.jmaa.2012.02.004
[28] Kokonendji, On d-orthogonality of the Sheffer systems associated to a convolution semigroup, J. Comput. Appl. Math. 181 pp 83– (2005) · Zbl 1082.60008 · doi:10.1016/j.cam.2004.11.019
[29] Kokonendji, Characterizations of some polynomial variance functions by d-pseudo-orthogonality, J. Appl. Math. Comput. 19 pp 427– (2005) · Zbl 1076.62010 · doi:10.1007/BF02935816
[30] Lee, Division problem of moment functionals, Rocky Mt. J. Math. 32 (2) pp 739– (2002) · Zbl 1043.42016 · doi:10.1216/rmjm/1030539695
[31] Lamiri, d-orthogonality of Hermite type polynomials, Appl. Math. Comput. 202 pp 24– (2008) · Zbl 1154.33004 · doi:10.1016/j.amc.2007.11.040
[32] Lamiri, d-orthogonality of Humbert and Jacobi type polynomials, J. Math. Anal. Appl. 341 pp 24– (2008) · Zbl 1218.33011 · doi:10.1016/j.jmaa.2007.09.047
[33] Marcellán, On the solution of some distributional differential equations: Existence and characterizations of the classical moment functionals, Integral Transforms Spec. Funct. 2 pp 185– (1994) · Zbl 0832.33006 · doi:10.1080/10652469408819050
[34] Marcellán, Second structure relation for semi-classical orthogonal polynomials, J. Comput. Appl. Math. 200 pp 537– (2007) · Zbl 1125.33008 · doi:10.1016/j.cam.2006.01.007
[35] Marcellán, Inverse finite-type relations between sequences of polynomials, Rev. Acad. Colomb. Cienc. XXXII 123 pp 245– (2008) · Zbl 1173.42012
[36] Maroni, Prolégomènes à l’étude des polynômes orthogonaux semi-classiques, Ann. Mat. Pura Appl. 149 pp 165– (1987) · Zbl 0636.33009 · doi:10.1007/BF01773932
[37] Maroni, L’orthogonalité et les récurrences de polynômes d’ordre supérieur à deux, Ann. Fac. Sci. Toulouse 10 pp 105– (1989) · Zbl 0707.42019 · doi:10.5802/afst.672
[38] Maroni, Orthogonal Polynomials and their Applications, IMACS Ann. Comput. Appl. Math pp 95– (1991)
[39] Maroni, Two-dimentional orthogonal polynomials, their associated sets and co-recursive sets, Numer. Algorithms 3 pp 299– (1992) · Zbl 0779.42013 · doi:10.1007/BF02141938
[40] Maroni, Variations around classical orthogonal polynomials. Connected problems, J. Comput. Appl. Math. 48 pp 133– (1993) · Zbl 0790.33006 · doi:10.1016/0377-0427(93)90319-7
[41] Maroni, Semi-classical character and finite-type relation between polynomial sequences, Appl. Numer. Math. 31 pp 295– (1999) · Zbl 0962.42017 · doi:10.1016/S0168-9274(98)00137-8
[42] Maroni, Diagonal orthogonal polynomial sequences, Methods Appl. Anal. 7 pp 769– (2000) · Zbl 1025.42013
[43] Matos, Frobenius-Padé approximants for d-orthogonal series: Theory and computational aspects, App. Numer. Math. 52 pp 89– (2005) · Zbl 1059.41005 · doi:10.1016/j.apnum.2004.03.003
[44] Petronilho, On the linear functionals associated to linearly related sequences of orthogonal polynomials, J. Math. Anal. Appl. 315 pp 379– (2006) · Zbl 1084.42020 · doi:10.1016/j.jmaa.2005.05.018
[45] Saib, Some inverse problems of d-orthogonal polynomials, Mediterr. J. Math. 10 pp 865– (2013) · Zbl 1267.42030 · doi:10.1007/s00009-012-0225-1
[46] G. Szegö Orthogonal polynomials. Amer. Math. Soc. Colloq. Publ. Vol. 23 (Amer. Math. Soc., Providence, RI 1975 Fourth Edition
[47] Vinet, Automorphisms of the Heisenberg-Weyl algebra and d-orthogonal polynomials, J. Math. Phys. 50 pp 033511– (2009) · Zbl 1202.33018 · doi:10.1063/1.3087425
[48] Vinet, d-orthogonal Charlier Polynomials and the Weyl Algebra, J. Phys. Conf. Ser. 284 pp 012060– (2011) · doi:10.1088/1742-6596/284/1/012060
[49] Zaghouani, Some basic d-orthogonal polynomial sets, Georgian Math. J. 12 pp 583– (2005) · Zbl 1091.42020
[50] Zerouki, On the 2-orthogonal polynomials and the generalized birth and death processes, Int. J. Math. Math. Sci. pp 1– (2006) · Zbl 1129.60077
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.