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Multiple orthogonal polynomials for classical weights. (English) Zbl 1033.33002
A new set of special functions, which has a wide range of applications from number theory to integrability of nonlinear dynamical systems, is described. The multiple orthogonal polynomials with respect to \(p>1\) weights satisfying Pearson’s equation. In particular, a classification of multiple orthogonal polynomials with respect to classical weights, which is based on properties of the corresponding Rodrigues operators, is given. It is shown that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order \(p+1\). The explicit formulas and recurrence relations for these polynomials are also obtained.

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
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[1] A. I. Aptekarev, Multiple orthogonal polynomials, Proceedings of the VIIIth Symposium on Orthogonal Polynomials and Their Applications (Seville, 1997), 1998, pp. 423 – 447. · Zbl 0958.42015
[2] 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. · Zbl 0917.47028
[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), no. 4, 487 – 524. · Zbl 0966.41009
[4] 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. · Zbl 0878.33004
[5] F. Beukers, A note on the irrationality of \?(2) and \?(3), Bull. London Math. Soc. 11 (1979), no. 3, 268 – 272. · Zbl 0421.10023
[6] Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. · Zbl 0840.26002
[7] Amílcar Branquinho, A note on semi-classical orthogonal polynomials, Bull. Belg. Math. Soc. Simon Stevin 3 (1996), no. 1, 1 – 12. · Zbl 0862.42018
[8] F. Marcellán, A. Branquinho, and J. Petronilho, Classical orthogonal polynomials: a functional approach, Acta Appl. Math. 34 (1994), no. 3, 283 – 303. · Zbl 0793.33009
[9] 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. · Zbl 0840.47026
[10] 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. · Zbl 0853.44006
[11] 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).
[12] 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. · Zbl 0733.41001
[13] 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. · Zbl 0671.33009
[14] 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). · Zbl 0754.41011
[15] V. N. Sorokin, Hermite-Padé approximants for Nikishin systems and the irrationality of \?(3), Uspekhi Mat. Nauk 49 (1994), no. 2(296), 167 – 168 (Russian); English transl., Russian Math. Surveys 49 (1994), no. 2, 176 – 177.
[16] 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. · Zbl 0952.42014
[17] 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. · Zbl 0962.42016
[18] 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. · Zbl 0969.33005
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