Geodesic
Difference equations having bases with powerlike growth which are perturbed by a spectral parameter
Sbornik. Mathematics, Tome 200 (2009) no. 5, pp. 753-781 Cet article a éte moissonné depuis la source Math-Net.Ru

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The asymptotic behaviour of solutions with powerlike growth of recurrence relations with a spectral parameter is investigated. A class of recurrence relations in which all basis solutions have powerlike growth is introduced. Recurrence relations in this class are linearly perturbed by a spectral parameter; for solutions of the new recurrence relations asymptotic formulae are obtained which are uniform with respect to the spectral parameter ranging within appropriate bounds. The theorems obtained are used for deriving new local asymptotic formulae for orthogonal and multiple orthogonal polynomials in a neighbourhood of the end-points of the support of the orthogonality weights. Bibliography: 14 titles.
Keywords: asymptotic behaviour of solutions of recurrence relations, local asymptotics of orthogonal and multiple orthogonal polynomials, Poincaré's theorem, Perron's theorem
Mots-clés : Birkhoff-Trjitzinsky theorem. 
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D. N. Tulyakov. Difference equations having bases with powerlike growth which are perturbed by a spectral parameter. Sbornik. Mathematics, Tome 200 (2009) no. 5, pp. 753-781. http://geodesic.mathdoc.fr/item/SM_2009_200_5_a6/

[1] H. Poincaré, “Sur les Equations Linéaires aux Différentielles Ordinaires et aux Différences Finies”, Amer. J. Math., 7:3 (1885), 203–258 | DOI | MR | Zbl

[2] O. Perron, “Über einen Satz des Herrn Poincaré”, J. Reine Angew. Math., 136 (1909), 17–37 | Zbl

[3] O. Perron, “Über die Poincarésche lineare Differenrengleichung”, J. Reine Angew. Math., 137 (1910), 6–64 | Zbl

[4] A. O. Gel'fond, Differenzenrechnung, VEB, Berlin, 1958 | MR | MR | Zbl | Zbl

[5] D. G. Birkhoff, “General theory of linear difference equations”, Trans. Amer. Math. Soc., 12:2 (1911), 243–284 | DOI | MR | Zbl

[6] D. G. Birkhoff, W. J. Trjitzinsky, “Analytic theory of singular difference equations”, Acta Math., 60:1 (1933), 1–89 | DOI | MR | Zbl

[7] D. G. Birkhoff, “Formal theory of irregular linear difference equations”, Acta Math., 54:1 (1930), 205–246 | DOI | MR | Zbl

[8] A. I. Aptekarev, “Asymptotics of orthogonal polynomials in a neighborhood of the endpoints of the interval of orthogonality”, Russian Acad. Sci. Sb. Math., 76:1 (1993), 35–50 | DOI | MR | Zbl | Zbl

[9] D. N. Tulyakov, “Local asymptotics of the ratio of orthogonal polynomials in the neighbourhood of an end-point of the support of the orthogonality measure”, Sb. Math., 192:2 (2001), 299–321 | DOI | MR | Zbl

[10] S. P. Suetin, “O silnoi asimptotike mnogochlenov, ortogonalnykh otnositelno kompleksnogo vesa”, Matem. sb., 200:1 (2009), 81–96

[11] V. A. Kaljagin, “On a class of polynomials defined by two orthogonality relations”, Math. USSR-Sb., 38:4 (1981), 563–580 | DOI | MR | Zbl

[12] V. N. Sorokin, “A generalization of classical orthogonal polynomials and the convergence of simultaneous Pade approximants”, J. Soviet Math., 45:6 (1989), 1461–1499 | DOI | MR | Zbl

[13] N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner, New York; Oliver Boyd, Edinburgh–London, 1965 | MR | MR | Zbl | Zbl

[14] V. Kaliaguinea, A. Ronveaux, “On an system of “classical” polynomials of simultaneous orthogonality”, J. Comput. Appl. Math., 67:2 (1996), 207–217 | DOI | MR | Zbl