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**, 237-249

MSC (2000):
Primary 11J72; Secondary 33C45, 41A21

DOI:
https://doi.org/10.1090/S0077-1554-09-00175-7

Published electronically:
December 3, 2009

MathSciNet review:
2573642

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**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, 3887–3914. MR**1990569**, https://doi.org/10.1090/S0002-9947-03-03330-0**3.**D. V. Khristoforov,*Recurrence relations for the Hermite-Padé approximants of a system of four functions of Markov and Stieltjes type*, in [**1**], pp. 11-26. (Russian)**4.**A. I. Bogolyubskii,*Recurrence relations with rational coefficients for some jointly orthogonal polynomials defined by Rodrigues' formula*, in [**1**], pp. 27-35. (Russian)**5.**A. I. Aptekarev and D. N. Tulyakov,*Four-term recurrence relations for -forms*, in [**1**]. pp. 37-43. (Russian)**6.**A. I. Aptekarev and V. G. Lysov,*Asymptotics of -forms jointly generated by orthogonal polynomials*, in [**1**]. pp. 55-62. (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://www-fourier.ujf-grenoble.fr/rivoal/articles.html.**9.**A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász,*Factoring polynomials with rational coefficients*, Math. Ann.**261**(1982), no. 4, 515–534. MR**682664**, https://doi.org/10.1007/BF01457454

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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:
https://doi.org/10.1090/S0077-1554-09-00175-7

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