On the determinant of an integral lattice generated by rational approximants of the Euler constant
Authors:
A. I. Aptekarev and D. N. Tulyakov
Translated by:
Alex Martsinkovsky
Original publication:
Trudy Moskovskogo Matematicheskogo Obshchestva, tom 70 (2009).
Journal:
Trans. Moscow Math. Soc. 2009, 237249
MSC (2000):
Primary 11J72; Secondary 33C45, 41A21
Published electronically:
December 3, 2009
MathSciNet review:
2573642
Fulltext PDF Free Access
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Abstract: We investigate rational approximants of the Euler constant constructed using a certain system of jointly orthogonal polynomials and ``averaging'' such approximants of mediant type. The properties of such approximants are related to the properties of an integral lattice in constructed from recurrently generated sequences. We also obtain estimates on the metric properties of a reduced basis, which imply that the forms with coefficients constructed from basis vectors of the lattice tend to zero. The question of whether the Euler constant is irrational is reduced to a property of the bases of lattices.
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A.
I. Aptekarev, A.
Branquinho, and W.
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(2004g:33014), http://dx.doi.org/10.1090/S0002994703033300
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A. I. Aptekarev and D. N. Tulyakov, Fourterm recurrence relations for forms, in [1]. pp. 3743. (Russian)
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A. I. Aptekarev and V. G. Lysov, Asymptotics of forms jointly generated by orthogonal polynomials, in [1]. pp. 5562. (Russian)
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D. N. Tulyakov, A system of recurrence relations for rational approximants of the Euler constant, Mat. Zametki (to appear).
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T. Rivoal, Rational approximations for values of derivatives of the gamma function, http://wwwfourier.ujfgrenoble.fr/rivoal/articles.html.
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 1.
 A. I. Aptekarev (ed.), Rational approximation of Euler's constant and recurrence relations, Current Problems in Math., vol. 9, Steklov Math. Inst. RAN, 2007. (Russian)
 2.
 A. I. Aptekarev, A. Branquinho, and W. Van Assche, Multiple orthogonal polynomials for classical weights, Trans. Amer. Math. Soc. 355 (2003), no. 10, 38873914. MR 1990569 (2004g:33014)
 3.
 D. V. Khristoforov, Recurrence relations for the HermitePadé approximants of a system of four functions of Markov and Stieltjes type, in [1], pp. 1126. (Russian)
 4.
 A. I. Bogolyubskii, Recurrence relations with rational coefficients for some jointly orthogonal polynomials defined by Rodrigues' formula, in [1], pp. 2735. (Russian)
 5.
 A. I. Aptekarev and D. N. Tulyakov, Fourterm recurrence relations for forms, in [1]. pp. 3743. (Russian)
 6.
 A. I. Aptekarev and V. G. Lysov, Asymptotics of forms jointly generated by orthogonal polynomials, in [1]. pp. 5562. (Russian)
 7.
 D. N. Tulyakov, A system of recurrence relations for rational approximants of the Euler constant, Mat. Zametki (to appear).
 8.
 T. Rivoal, Rational approximations for values of derivatives of the gamma function, http://wwwfourier.ujfgrenoble.fr/rivoal/articles.html.
 9.
 A. K. Lenstra, H. W. Lenstra, and L. Lovasz, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515534. MR 682664 (84a:12002)
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Additional Information
A. I. Aptekarev
Affiliation:
M. V. Keldysh Institute of Applied Mathematics, 4 Miusskaya Ploshchad′, Moscow 125047, Russia, and Lomonosov State University, Moscow, Russia
Email:
aptekaa@keldysh.ru
D. N. Tulyakov
Affiliation:
M. V. Keldysh Institute of Applied Mathematics, 4 Miusskaya Ploshchad′, Moscow 125047, Russia
DOI:
http://dx.doi.org/10.1090/S0077155409001757
PII:
S 00771554(09)001757
Published electronically:
December 3, 2009
Additional Notes:
Partially supported by RFFI, Project No. 08–01–00179, Program No. 1 OMN RAN, and the Support Program for Leading Scientific Schools (Project NSh–3906.2008.1).
Article copyright:
© Copyright 2009
American Mathematical Society
