Geodesic
Zeros and asymptotics of polynomials satisfying three-term recurrence relations with complex coefficients
Sbornik. Mathematics, Tome 80 (1995) no. 2, pp. 309-333 
D. Barrios; G. L. Lopes; E. Torrano. Zeros and asymptotics of polynomials satisfying three-term recurrence relations with complex coefficients. Sbornik. Mathematics, Tome 80 (1995) no. 2, pp. 309-333. http://geodesic.mathdoc.fr/item/SM_1995_80_2_a3/
@article{SM_1995_80_2_a3,
     author = {D. Barrios and G. L. Lopes and E. Torrano},
     title = {Zeros and asymptotics of polynomials satisfying three-term recurrence relations with complex coefficients},
     journal = {Sbornik. Mathematics},
     pages = {309--333},
     year = {1995},
     volume = {80},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1995_80_2_a3/}
}
TY  - JOUR
AU  - D. Barrios
AU  - G. L. Lopes
AU  - E. Torrano
TI  - Zeros and asymptotics of polynomials satisfying three-term recurrence relations with complex coefficients
JO  - Sbornik. Mathematics
PY  - 1995
SP  - 309
EP  - 333
VL  - 80
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1995_80_2_a3/
LA  - en
ID  - SM_1995_80_2_a3
ER  - 
%0 Journal Article
%A D. Barrios
%A G. L. Lopes
%A E. Torrano
%T Zeros and asymptotics of polynomials satisfying three-term recurrence relations with complex coefficients
%J Sbornik. Mathematics
%D 1995
%P 309-333
%V 80
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1995_80_2_a3/
%G en
%F SM_1995_80_2_a3

Voir la notice de l'article provenant de la source Math-Net.Ru

Under very general conditions on the complex coefficients of a three-term recurrence relation, it is proved that 'almost all' zeros of the polynomials generated by these relations 'accumulate' on a certain segment in the complex plane. From this result follow the convergence of diagonal Padé approximants and a generalization of Van Vleck's theorem on the convergence of $S$-fractions. Another interesting application is an extension of the so-called Nevai–Blumenthal class of polynomials $M(a,2b)$ to the case when $a,b\in{\mathbb C}$.

[1] Gelfond A. O., Ischislenie konechnykh raznostei, 3-e izdanie, Nauka, M., 1967 | MR

[2] Gohberg Israel, Goldberg Seymour, Kaashoek Marinus A., Classes of Linear Operators, V. I, Birkhauser, Berlin, 1990 | MR

[3] Gonchar A. A., “O skhodimosti approksimatsii Pade dlya nekotorykh klassov meromorfnykh funktsii”, Matem. sb., 97(139):4 (1975), 607–629 | MR | Zbl

[4] Gonchar A. A., Lungu K. I., “Polyusy diagonalnykh approksimatsii Pade i analiticheskoe prodolzhenie funktsii”, Matem. sb., 111(153):2 (1980), 279–292 | MR | Zbl

[5] Gonchar A. A., “O ravnomernoi skhodimosti diagonalnykh approksimatsii Pade”, Matem. sb., 118(160):4(8) (1982), 535–556 | MR | Zbl

[6] Gonchar A. A., “Polyusy strok tablitsy Pade i meromorfnoe prodolzhenie funktsii”, Matem. sb., 115(157):4 (1981), 590–613 | MR | Zbl

[7] Halmos Paul R., A Hilbert Space Problem Book, Springer-Verlag, New York, 1984 | MR

[8] Kato T., Perturbation Theory for Linear Operators, Springer, New York, 1976 | MR

[9] Lopes G., “Skhodimost approksimatsii Pade meromorfnykh funktsii stiltesovskogo tipa i sravnitelnaya asimptotika dlya ortogonalnykh mnogochlenov”, Matem. sb., 136(178):2(6) (1988), 206–226 | MR | Zbl

[10] López G. L., Vavilov V. V., “Survey on recent advances in inverse problems of Padé approximation theory”, Proc. of the Conf. on Rat. Approx. and Int. Tampa'83, L. Notes, 1105, Springer Verlag, Berlin, 1984

[11] Magnus A., “Toeplitz matrix techniques and convergence of complex weight Padé approximants”, Journal of Computational and Applied Mathematics, 19 (1987), 23–28 | MR

[12] Nevai P., “Orthogonal Polynomials”, Memoirs of the A.M.S., 18, no. 213, Rhode Island, 1979 | MR

[13] Rakhmanov E. A., “O skhodimosti diagonalnykh approksimatsii Pade”, Matem. sb., 104(146):2 (1977), 271–291 | MR | Zbl

[14] Wall H. S., Analytic Theory of Continued Fractions, Chelsea Pub. Company, N.Y., 1973 | MR