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

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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 $\mathbb R^3$ constructed from recurrently generated sequences. We also obtain estimates on the metric properties of a reduced basis, which imply that the $\gamma$-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.

- 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) - 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**, DOI https://doi.org/10.1090/S0002-9947-03-03330-0 - D. V. Khristoforov,
*Recurrence relations for the Hermite–Padé approximants of a system of four functions of Markov and Stieltjes type*, in \cite{A1}, pp. 11–26. (Russian) - A. I. Bogolyubskii,
*Recurrence relations with rational coefficients for some jointly orthogonal polynomials defined by Rodrigues’ formula*, in \cite{A1}, pp. 27–35. (Russian) - A. I. Aptekarev and D. N. Tulyakov,
*Four-term recurrence relations for $\gamma$-forms*, in \cite{A1}. pp. 37–43. (Russian) - A. I. Aptekarev and V. G. Lysov,
*Asymptotics of $\gamma$-forms jointly generated by orthogonal polynomials*, in \cite{A1}. pp. 55–62. (Russian) - D. N. Tulyakov,
*A system of recurrence relations for rational approximants of the Euler constant*, Mat. Zametki (to appear). - T. Rivoal,
*Rational approximations for values of derivatives of the gamma function*, http://www-fourier.ujf-grenoble.fr/rivoal/articles.html. - 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**, DOI 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

MR Author ID:
192572

Email:
aptekaa@keldysh.ru

**D. N. Tulyakov**

Affiliation:
M. V. Keldysh Institute of Applied Mathematics, 4 Miusskaya Ploshchad′, Moscow 125047, Russia

MR Author ID:
632175

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