Table of Contents
ISRN Mathematical Analysis
Volume 2011 (2011), Article ID 908508, 14 pages
http://dx.doi.org/10.5402/2011/908508
Research Article

Rational Divide-and-Conquer Relations

1Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathumthani 12120, Thailand
2Department of Mathematics, Faculty of Science, Kasetsart University, and Centre of Excellence in Mathematics, CHE, Mahidol University, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 28 October 2010; Accepted 12 December 2010

Academic Editor: G. L. Karakostas

Copyright Β© 2011 Charinthip Hengkrawit et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A rational divide-and-conquer relation, which is a natural generalization of the classical divide-and-conquer relation, is a recursive equation of the form 𝑓(𝑏𝑛)=𝑅(𝑓(𝑛),𝑓(𝑛),…,𝑓(π‘βˆ’1)𝑛)+𝑔(𝑛), where 𝑏 is a positive integer β‰₯2; 𝑅 a rational function in π‘βˆ’1 variables and 𝑔 a given function. Closed-form solutions of certain rational divide-and-conquer relations which can be used to characterize the trigonometric cotangent-tangent and the hyperbolic cotangent-tangent function solutions are derived and their global behaviors are investigated.