American Institute of Mathematical Sciences

Advanced
September  2004, 3(3): 447-464. doi: 10.3934/cpaa.2004.3.447

Upper and lower bounds on Mathieu characteristic numbers of integer orders

Armando G. M. Neves 1,

1. 

Departamento de Matemática, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627 - Caixa Postal 702, 30123-970 - B. Horizonte - MG, Brazil

Received  March 2003 Revised  March 2004 Published  June 2004

For each Mathieu characteristic number of integer order (MCN) we construct sequences of upper and lower bounds both converging to the MCN. The bounds arise as zeros of polynomials in sequences generated by recursion. This result is based on a constructive proof of convergence for Ince's continued fractions. An important role is also played by the fact that the continued fractions define meromorphic functions.
Keywords: Mathieu characteristic numbers, continued fractions, meromorphic functions..
Mathematics Subject Classification: 40A15, 30B70, 33E1.
Citation: Armando G. M. Neves. Upper and lower bounds on Mathieu characteristic numbers of integer orders. Communications on Pure & Applied Analysis, 2004, 3 (3) : 447-464. doi: 10.3934/cpaa.2004.3.447
[1]

Laura Luzzi, Stefano Marmi. On the entropy of Japanese continued fractions. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 673-711. doi: 10.3934/dcds.2008.20.673
[2]

Pierre Arnoux, Thomas A. Schmidt. Commensurable continued fractions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4389-4418. doi: 10.3934/dcds.2014.34.4389
[3]

Claudio Bonanno, Carlo Carminati, Stefano Isola, Giulio Tiozzo. Dynamics of continued fractions and kneading sequences of unimodal maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1313-1332. doi: 10.3934/dcds.2013.33.1313
[4]

Élise Janvresse, Benoît Rittaud, Thierry de la Rue. Dynamics of $\lambda$-continued fractions and $\beta$-shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1477-1498. doi: 10.3934/dcds.2013.33.1477
[5]

Jian-Hua Zheng. Dynamics of hyperbolic meromorphic functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2273-2298. doi: 10.3934/dcds.2015.35.2273
[6]

Zuxing Xuan. On conformal measures of parabolic meromorphic functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 249-257. doi: 10.3934/dcdsb.2015.20.249
[7]

Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2375-2393. doi: 10.3934/dcds.2018098
[8]

Marc Kessböhmer, Bernd O. Stratmann. On the asymptotic behaviour of the Lebesgue measure of sum-level sets for continued fractions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2437-2451. doi: 10.3934/dcds.2012.32.2437
[9]

Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417
[10]

Agnieszka Badeńska. Measure rigidity for some transcendental meromorphic functions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2375-2402. doi: 10.3934/dcds.2012.32.2375
[11]

Kanji Inui, Hikaru Okada, Hiroki Sumi. The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 753-766. doi: 10.3934/dcds.2020060
[12]

Shahar Nevo, Xuecheng Pang and Lawrence Zalcman. Picard-Hayman behavior of derivatives of meromorphic functions with multiple zeros. Electronic Research Announcements, 2006, 12: 37-43.

[13]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41
[14]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75
[15]

Robert F. Bailey, John N. Bray. Decoding the Mathieu group M12. Advances in Mathematics of Communications, 2007, 1 (4) : 477-487. doi: 10.3934/amc.2007.1.477
[16]

Peng Sun. Exponential decay of Lebesgue numbers. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3773-3785. doi: 10.3934/dcds.2012.32.3773
[17]

Danny Calegari, Alden Walker. Ziggurats and rotation numbers. Journal of Modern Dynamics, 2011, 5 (4) : 711-746. doi: 10.3934/jmd.2011.5.711
[18]

Xavier Buff, Nataliya Goncharuk. Complex rotation numbers. Journal of Modern Dynamics, 2015, 9: 169-190. doi: 10.3934/jmd.2015.9.169
[19]

Vittorio Martino. On the characteristic curvature operator. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1911-1922. doi: 10.3934/cpaa.2012.11.1911
[20]

Michael Björklund, Alexander Gorodnik. Central limit theorems in the geometry of numbers. Electronic Research Announcements, 2017, 24: 110-122. doi: 10.3934/era.2017.24.012

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences