Asymptotics of quadratic Hermite--Pad\'e approximants of the exponential functions
Problemy fiziki, matematiki i tehniki, no. 1 (2014), pp. 74-80.

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The paper deals with asymptotic properties of diagonal quadratic Hermite–Padé approximants of type I for exponential system $\{e^{\lambda_jz}\}_{j=0}^2$ with arbitrary real $\lambda_0$, $\lambda_1$, $\lambda_2$. The proved theorems complement the known results of P. Borwein, F. Wielonsky.
Mots-clés : Hermite–Padé approximants of type I
Keywords: quadratic and Hermite–Padé approximants, asymptotic equality, saddle-point method.
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A. P. Starovoitov. Asymptotics of quadratic Hermite--Pad\'e approximants of the exponential functions. Problemy fiziki, matematiki i tehniki, no. 1 (2014), pp. 74-80. http://geodesic.mathdoc.fr/item/PFMT_2014_1_a12/

[1] K. Mahler, “Perfect systems”, Comp. Math., 19 (1968), 95–166 | MR | Zbl

[2] C. Hermite, “Sur la fonction exponentielle”, C.R. Akad. Sci. (Paris), 77 (1873), 18–293

[3] F. Klein, Elementarnaya matematika s tochki zreniya vysshei, v. 1, Nauka, M., 1933

[4] C. Hermite, “Sur la généralisation des fractions continues algébriques”, Ann. Math. Pura. Appl. Ser. 2A, 21 (1883), 289–308

[5] K. Mahler, “Zur Approximation der Exponentialfunktion und des Logarithmus. I; II”, J. Reine Angew. Math., 166 (1931), 118–150

[6] H. Pade, “Memoire sur les developpements en fractions continues de la fonctial exponential”, Ann. École Norm. Sup. Paris, 16:3 (1899), 394–426 | MR

[7] A. I. Aptekarev, H. Stahl, “Asymptotics of Hermite–Pade polynomials”, Progress in Approximation Theory, eds. A. A. Gonchar, E. B. Saff, Springer-Verlag, New York–Berlin, 1992, 127–167 | DOI | MR | Zbl

[8] A. I. Aptekarev i dr., “Approksimatsii Pade, nepreryvnye drobi i ortogonalnye mnogochleny”, Uspekhi matem. nauk, 66:6(402) (2011), 37–122 | DOI | MR | Zbl

[9] K. Mahler, “Applications of some formulas by Hermite to the approximation of exponentials and logarithms”, Math. Ann., 1668 (1967), 200–227 | DOI | MR

[10] G. V. Chudnovsky, “Hermite–Pade approximations to exponential functions and elementary estimates of the measure of irrationality of $\pi$”, Lecture Notes in Math., 925, Springer-Verlag, New York–Berlin, 1982, 299–322 | DOI | MR

[11] A. I. Aptekarev, “Ratsionalnye priblizheniya postoyannoi Eilera i rekurrentnye sootnosheniya”, Sbornik statei, Sovr. probl. matem., 9, ed. A. I. Aptekarev, MIAN, M., 1988, 1–71

[12] V. A. Kalyagin, “Approksimatsii Ermita–Pade i spektralnyi analiz nesimmetrichnykh operatorov”, Matem. sbornik, 185:6 (1994), 79–100 | MR | Zbl

[13] S. P. Suetin, “Approksimatsii Pade i effektivnoe analiticheskoe prodolzhenie stepennogo ryada”, Uspekhi matem. nauk, 57:1 (2002), 45–142 | DOI | MR | Zbl

[14] P. B. Borwein, “Quadratic Hermite–Padé approximation to the exponential function”, Const. Approx., 62 (1986), 291–302 | DOI | MR

[15] F. Wielonsky, “Asymptotics of Diagonal Hermite–Padé Approximants to $e^z$”, J. Approx. Theory, 90:2 (1997), 283–298 | DOI | MR | Zbl

[16] A. I. Aptekarev, “O skhodimosti ratsionalnykh approksimatsii k naboru eksponent”, Vestn. MGU. Ser. 1. Matematika. Mekhanika, 1981, no. 1, 68–74 | MR | Zbl

[17] A. P. Starovoitov, “Asimptotika ermitovoi approksimatsii eksponent”, Izvestiya Gomelskogo gosudarstvennogo universiteta im. F. Skoriny, 2012, no. 5(74), 163–171 | MR

[18] A. P. Starovoitov, “Approksimatsii Ermita–Pade dlya sistemy funktsii Mittag–Lefflera”, Problemy fiziki, matematiki i tekhniki, 2013, no. 1(14), 81–87

[19] Yu. V. Sidorov, M. V. Fedoryuk, M. I. Shabunin, Lektsii po teorii funktsii kompleksnogo peremennogo, Nauka, M., 1989 | MR | Zbl

[20] A. I. Markushevich, Teoriya analiticheskikh funktsii, v. 1, Nauka, M., 1967