Multiple orthogonal polynomials for classical weights

Aptekarev, A.; Branquinho, A.; Van Assche, W.

doi:10.1090/S0002-9947-03-03330-0

Multiple orthogonal polynomials for classical weights
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by A. I. Aptekarev, A. Branquinho and W. Van Assche PDF

Trans. Amer. Math. Soc. 355 (2003), 3887-3914 Request permission

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

A. I. Aptekarev, Multiple orthogonal polynomials, Proceedings of the VIIIth Symposium on Orthogonal Polynomials and Their Applications (Seville, 1997), 1998, pp. 423–447. MR 1662713, DOI 10.1016/S0377-0427(98)00175-7
A. Aptekarev and V. Kaliaguine, Complex rational approximation and difference operators, Proceedings of the Third International Conference on Functional Analysis and Approximation Theory, Vol. I (Acquafredda di Maratea, 1996), 1998, pp. 3–21. MR 1644539
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), no. 4, 487–524. MR 1771693, DOI 10.1007/s003650010004
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. MR 1458825, DOI 10.1006/jath.1996.3074
F. Beukers, A note on the irrationality of $\zeta (2)$ and $\zeta (3)$, Bull. London Math. Soc. 11 (1979), no. 3, 268–272. MR 554391, DOI 10.1112/blms/11.3.268
Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960, DOI 10.1007/978-1-4612-0793-1
Amílcar Branquinho, A note on semi-classical orthogonal polynomials, Bull. Belg. Math. Soc. Simon Stevin 3 (1996), no. 1, 1–12. MR 1378494
F. Marcellán, A. Branquinho, and J. Petronilho, Classical orthogonal polynomials: a functional approach, Acta Appl. Math. 34 (1994), no. 3, 283–303. MR 1273613, DOI 10.1007/BF00998681
V. A. Kalyagin, Hermite-Padé approximants and spectral analysis of nonsymmetric operators, Mat. Sb. 185 (1994), no. 6, 79–100 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 82 (1995), no. 1, 199–216. MR 1280397, DOI 10.1070/SM1995v082n01ABEH003558
V. Kaliaguine, The operator moment problem, vector continued fractions and an explicit form of the Favard theorem for vector orthogonal polynomials, Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994), 1995, pp. 181–193. MR 1379130, DOI 10.1016/0377-0427(95)00109-3
L. R. Pineĭro Dias, On simultaneous approximations for some collection of Markov functions, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2 (1987), 67–70, 103 (Russian). MR 884516
E. M. Nikishin and V. N. Sorokin, Rational approximations and orthogonality, Translations of Mathematical Monographs, vol. 92, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by Ralph P. Boas. MR 1130396, DOI 10.1090/mmono/092
V. N. Sorokin, Generalization of classical orthogonal polynomials and convergence of simultaneous Padé approximants, Trudy Sem. Petrovsk. 11 (1986), 125–165, 245, 247 (Russian, with English summary); English transl., J. Soviet Math. 45 (1989), no. 6, 1461–1499. MR 834171, DOI 10.1007/BF01097274
V. N. Sorokin, Simultaneous Padé approximations of functions of Stieltjes type, Sibirsk. Mat. Zh. 31 (1990), no. 5, 128–137, 215 (Russian); English transl., Siberian Math. J. 31 (1990), no. 5, 809–817 (1991). MR 1088923, DOI 10.1007/BF00974495
V. N. Sorokin, Hermite-Padé approximants for Nikishin systems and the irrationality of $\zeta (3)$, Uspekhi Mat. Nauk 49 (1994), no. 2(296), 167–168 (Russian); English transl., Russian Math. Surveys 49 (1994), no. 2, 176–177. MR 1283150, DOI 10.1070/RM1994v049n02ABEH002229
Walter Van Assche, Multiple orthogonal polynomials, irrationality and transcendence, Continued fractions: from analytic number theory to constructive approximation (Columbia, MO, 1998) Contemp. Math., vol. 236, Amer. Math. Soc., Providence, RI, 1999, pp. 325–342. MR 1665377, DOI 10.1090/conm/236/03504
Walter Van Assche, Nonsymmetric linear difference equations for multiple orthogonal polynomials, SIDE III—symmetries and integrability of difference equations (Sabaudia, 1998) CRM Proc. Lecture Notes, vol. 25, Amer. Math. Soc., Providence, RI, 2000, pp. 415–429. MR 1771743, DOI 10.1090/crmp/025/40
Walter Van Assche and Els Coussement, Some classical multiple orthogonal polynomials, J. Comput. Appl. Math. 127 (2001), no. 1-2, 317–347. Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials. MR 1808581, DOI 10.1016/S0377-0427(00)00503-3