Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter June 11, 2020

Asymptotic relations involving đť‘‘-orthogonal polynomials

  • Imed Lamiri EMAIL logo and Jihen Weslati

Abstract

In this paper, we consider a natural extension in the context of d-orthogonality for asymptotic analysis of orthogonal polynomials. We introduce, for several d-orthogonal polynomials, asymptotic expansions in terms of d-Hermite ones. From these expansions, several limits between d-orthogonal polynomials are obtained.

MSC 2010: 33C45; 42C05

References

[1] A. I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), no. 1–2, 423–447. 10.1016/S0377-0427(98)00175-7Search in Google Scholar

[2] A. I. Aptekarev, F. Marcellán and I. A. Rocha, Semiclassical multiple orthogonal polynomials and the properties of Jacobi–Bessel polynomials, J. Approx. Theory 90 (1997), no. 1, 117–146. 10.1006/jath.1996.3074Search in Google Scholar

[3] J. Arvesú, J. Coussement and W. Van Assche, Some discrete multiple orthogonal polynomials, J. Comput. Appl. Math. 153 (2003), no. 1–2, 19–45. 10.1016/S0377-0427(02)00597-6Search in Google Scholar

[4] Y. Ben Cheikh and K. Douak, On two-orthogonal polynomials related to the Bateman’s Jnu,v-function, Methods Appl. Anal. 7 (2000), no. 4, 641–662. 10.4310/MAA.2000.v7.n4.a3Search in Google Scholar

[5] Y. Ben Cheikh and I. Lamiri, Generating functions via integral transforms, J. Math. Anal. Appl. 331 (2007), no. 2, 1200–1229. 10.1016/j.jmaa.2006.09.052Search in Google Scholar

[6] Y. Ben Cheikh, I. Lamiri and A. Ouni, On Askey-scheme and d-orthogonality. I. A characterization theorem, J. Comput. Appl. Math. 233 (2009), no. 3, 621–629. 10.1016/j.cam.2009.02.029Search in Google Scholar

[7] Y. Ben Cheikh and A. Zaghouani, Some discrete d-orthogonal polynomial sets, J. Comput. Appl. Math. 156 (2003), no. 2, 253–263. 10.1016/S0377-0427(02)00914-7Search in Google Scholar

[8] F. Brafman, Some generating functions for Laguerre and Hermite polynomials, Canadian J. Math. 9 (1957), 180–187. 10.4153/CJM-1957-020-1Search in Google Scholar

[9] Y. B. Cheikh and A. Zaghouani, d-orthogonality via generating functions, J. Comput. Appl. Math. 199 (2007), no. 1, 2–22. 10.1016/j.cam.2005.01.051Search in Google Scholar

[10] T. S. Chihara, An Introduction to Orthogonal Polynomials, Math. Appl. 13, Gordon and Breach Science, New York, 1978, Search in Google Scholar

[11] M. G. de Bruin, Simultaneous Padé approximation and orthogonality, Orthogonal Polynomials and Applications (Bar-le-Duc 1984), Lecture Notes in Math. 1171, Springer, Berlin (1985), 74–83. 10.1007/BFb0076532Search in Google Scholar

[12] J. Devisme, Sur l’équation de M. Pierre Humbert, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (3) 25 (1933), 143–238. 10.5802/afst.380Search in Google Scholar

[13] K. Douak, The relation of the d-orthogonal polynomials to the Appell polynomials, J. Comput. Appl. Math. 70 (1996), no. 2, 279–295. 10.1016/0377-0427(95)00211-1Search in Google Scholar

[14] K. Douak and P. Maroni, On d-orthogonal Tchebychev polynomials. I, Appl. Numer. Math. 24 (1997), no. 1, 23–53. 10.1016/S0168-9274(97)00006-8Search in Google Scholar

[15] K. Douak and P. Maroni, On d-orthogonal Tchebychev polynomials. II, Methods Appl. Anal. 4 (1997), no. 4, 404–429. 10.4310/MAA.1997.v4.n4.a3Search in Google Scholar

[16] Z. Ebtissem and B. Ammar, On the 2-orthogonal polynomials and the generalized birth and death processes, Int. J. Math. Math. Sci. 2006 (2006), Article ID 28131. 10.1155/IJMMS/2006/28131Search in Google Scholar

[17] H. W. Gould and A. T. Hopper, Operational formulas connected with two generalizations of Hermite polynomials, Duke Math. J. 29 (1962), 51–63. 10.1215/S0012-7094-62-02907-1Search in Google Scholar

[18] P. Humbert, Some extensions of Pincherle’s polynomials, Proc. Edinb. Math. Soc. 39 (1920), 21–24. 10.1017/S0013091500035756Search in Google Scholar

[19] V. Kaliaguine and A. Ronveaux, On a system of “classical” polynomials of simultaneous orthogonality, J. Comput. Appl. Math. 67 (1996), no. 2, 207–217. 10.1016/0377-0427(94)00129-4Search in Google Scholar

[20] R. Koekoek and R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Report no. 98-17, Delft University of Technology, Delft, 1998. Search in Google Scholar

[21] I. Lamiri, d-orthogonality of discrete q-Hermite type polynomials, J. Approx. Theory 170 (2013), 116–133. 10.1016/j.jat.2012.07.002Search in Google Scholar

[22] I. Lamiri and A. Ouni, d-orthogonality of Hermite type polynomials, Appl. Math. Comput. 202 (2008), no. 1, 24–43. 10.1016/j.amc.2007.11.040Search in Google Scholar

[23] I. Lamiri and A. Ouni, d-orthogonality of Humbert and Jacobi type polynomials, J. Math. Anal. Appl. 341 (2008), no. 1, 24–51. 10.1016/j.jmaa.2007.09.047Search in Google Scholar

[24] J. L. López and N. M. Temme, Approximation of orthogonal polynomials in terms of Hermite polynomials, Methods Appl. Anal. 6 (1999), no. 2, 131–146. 10.4310/MAA.1999.v6.n2.a1Search in Google Scholar

[25] J. L. López and N. M. Temme, Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials, J. Math. Anal. Appl. 239 (1999), no. 2, 457–477. 10.1006/jmaa.1999.6584Search in Google Scholar

[26] P. Maroni, L’orthogonalité et les récurrences de polynômes d’ordre supérieur à deux, Ann. Fac. Sci. Toulouse Math. (5) 10 (1989), no. 1, 105–139. 10.5802/afst.672Search in Google Scholar

[27] G. V. Milovanović and G. B. Dordević, On some properties of Humbert’s polynomials. II, Facta Univ. Ser. Math. Inform. (1991), no. 6, 23–30. Search in Google Scholar

[28] S. Pincherle, Una nuova estensione delle funzioni sferiche (in Italian), Bologna Mem. (5) 1 (1890), 337–369. Search in Google Scholar

[29] G. Szegő, Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ. 23, American Mathematical Society, Providence, 1975. Search in Google Scholar

[30] N. M. Temme and J. L. López, The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis, J. Comput. Appl. Math. 133 (2001), no. 1–2, 623–633. 10.1016/S0377-0427(00)00683-XSearch in Google Scholar

[31] W. Van Assche and S. B. Yakubovich, Multiple orthogonal polynomials associated with Macdonald functions, Integral Transform. Spec. Funct. 9 (2000), no. 3, 229–244. 10.1080/10652460008819257Search in Google Scholar

[32] J. Van Iseghem, Vector orthogonal relations. Vector QD-algorithm, J. Comput. Appl. Math. 19 (1987), no. 1, 141–150. 10.1016/S0377-0427(87)80019-5Search in Google Scholar

Received: 2019-01-04
Accepted: 2019-02-26
Published Online: 2020-06-11
Published in Print: 2021-06-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 20.4.2024 from https://www.degruyter.com/document/doi/10.1515/gmj-2020-2063/html
Scroll to top button