Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Multiple orthogonal polynomials for classical weights

Authors: A. I. Aptekarev, A. Branquinho and W. Van Assche
Journal: Trans. Amer. Math. Soc. 355 (2003), 3887-3914
MSC (2000): Primary 33C45, 42C05
Published electronically: June 10, 2003
MathSciNet review: 1990569
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A new set of special functions, which has a wide range of applications from number theory to integrability of nonlinear dynamical systems, is described. We study multiple orthogonal polynomials with respect to $p > 1$ weights satisfying Pearson's equation. In particular, we give a classification of multiple orthogonal polynomials with respect to classical weights, which is based on properties of the corresponding Rodrigues operators. We show that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order $p+1$. We also obtain explicit formulas and recurrence relations for these polynomials.

References [Enhancements On Off] (What's this?)

  • 1. A. I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 423-447. MR 99m:42036
  • 2. A. I. Aptekarev and V. Kaliaguine, Complex rational approximation and difference operators, Rend. Circ. Matem. Palermo, Ser. II, suppl. 52 (1998), 3-21. MR 99h:41019
  • 3. A. Aptekarev, V. Kaliaguine, and J. Van Iseghem, The genetic sums' representation for the moments of a system of Stieltjes functions and its application, Constr. Approx. 16 (2000), 487-524. MR 2001g:41021
  • 4. A. I. Aptekarev, F. Marcellán, and I. Rocha, Semiclassical multiple orthogonal polynomials and the properties of Jacobi-Bessel polynomials, J. Approx. Theory 90 (1) (1997), 117-146. MR 98k:33012
  • 5. F. Beukers, A note on the irrationality of $\zeta(2)$ and $\zeta(3)$, Bull. London Math. Soc. 11 (1979), 268-272. MR 81j:10045
  • 6. P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Graduate Texts in Mathematics 161, Springer-Verlag, New York, 1995. MR 97e:41001
  • 7. A. Branquinho, A note on semi-classical orthogonal polynomials, Bull. Belg. Math. Soc. 3 (1996), 1-12. MR 97d:33001
  • 8. A. Branquinho, F. Marcellán, and J. Petronilho, Classical orthogonal polynomials: A functional approach, Acta Appl. Math. 34, no. 3 (1994), 283-303. MR 95b:33024
  • 9. V. Kalyagin, Hermite-Padé approximants and spectral analysis of non-symmetric operators, Mat. Sb. 185 (1994), 79-100; English translation in Russian Acad. Sci. Sb. Math. 82 (1995), 199-216. MR 95d:47038
  • 10. V. Kaliaguine, The operator moment problem, vector continued fractions and an explicit form of the Favard theorem for vector orthogonal polynomials, J. Comput. Appl. Math. 65 (1995), 181-193. MR 97c:44002
  • 11. L. R. Piñeiro, On simultaneous approximations for a collection of Markov functions, Vestnik Mosk. Univ., Ser. I (1987), no. 2, 67-70; English translation in Moscow Univ. Math. Bull. 42 (2) (1987), 52-55. MR 88c:41033
  • 12. E. M. Nikishin and V. N. Sorokin, Rational Approximations and Orthogonality, Amer. Math. Soc. Transl. (2), vol. 92, Amer. Math. Soc., Providence, Rhode Island, 1991. MR 92i:30037
  • 13. V. N. Sorokin, Generalization of classical polynomials and convergence of simultaneous Padé approximants, Trudy Sem. Petrovsk. 11 (1986), 125-165; English translation in J. Soviet Math. 45 (1986), 1461-1499. MR 87g:33011
  • 14. V. N. Sorokin, Simultaneous Padé approximation for functions of Stieltjes type, Sib. Mat. Zh. 31, no. 5 (1990), 128-137; English translation in Sib. Math. J. 31, no. 5 (1990), 809-817. MR 92f:41023
  • 15. V. N. Sorokin, Hermite-Padé approximations for Nikishin systems and the irrationality of $\zeta(3)$, Uspekhi Mat. Nauk 49, No. 2 (1994), 167-168; English translation in Russian Math. Surveys 49, No. 2 (1994), 176-177. MR 95c:11092
  • 16. W. Van Assche, Multiple orthogonal polynomials, irrationality and transcendence, in Continued Fractions: from analytic number theory to constructive approximation (B. C. Berndt, F. Gesztesy, eds.), Contemporary Mathematics 236 (1999), 325-342. MR 2000k:42039
  • 17. W. Van Assche, Non-symmetric linear difference equations for multiple orthogonal polynomials, CRM Proceedings and Lecture Notes, Vol. 25, Amer. Math. Soc., Providence, RI, 2000, pp. 391-405. MR 2001d:39010
  • 18. W. Van Assche and E. Coussement, Some classical multiple orthogonal polynomials, J. Comput. Appl. Math. 127 (2001), 317-347. MR 2001i:33012

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 33C45, 42C05

Retrieve articles in all journals with MSC (2000): 33C45, 42C05

Additional Information

A. I. Aptekarev
Affiliation: Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya Square 4, Moscow 125047, Russia

A. Branquinho
Affiliation: Departamento de Matemática da FCTUC, FCTUC, Universidade de Coimbra, Apartado 3008, 3000 Coimbra, Portugal

W. Van Assche
Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium

Received by editor(s): February 13, 2001
Published electronically: June 10, 2003
Additional Notes: This work was conducted in the framework of project INTAS-2000-272. The research was carried out while the first author was visiting the Universidade de Coimbra with a grant from Fundação para a Ciência e Tecnologia PRAXIS XXI/2654/98/BCC and the Katholieke Universiteit Leuven with senior fellowship F/99/009 of the research counsel. The first author is supported by grant RFBR 00-15-96132 and RFBR 02-01-00564. The second author is supported by Centro de Matemática da Universidade de Coimbra (CMUC). The third author is supported by grant G.0184.02 of FWO-Vlaanderen
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society