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Báo cáo toán học: "an Identity Generator: Basic Commutator"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: An Identity Generator: Basic Commutators. | An Identity Generator Basic Commutators M. Farrokhi D. G. Institute of Mathematics University of Tsukuba Tsukuba Ibaraki 305 Japan m.farrokhi.d.g@gmail.com Submitted Feb 23 2008 Accepted Apr 26 2008 Published May 5 2008 Mathematics Subject Classification Primary 05A19 68R15 Secondary 11B39 20E05 Abstract We introduce a group theoretical tool on which one can derive a family of identities from sequences that are defined by a recursive relation. As an illustration it is shown that n 1 X F n i F i 1 n i 1 where Fn denotes the sequence of Fibonacci numbers. 1 Preliminaries and Introduction We start our work with recalling some basic facts about the structural properties of words in a free group cf. 1 . Let F be the free group generated by the set X x1 . xng. Marshall Hall 1 introduced a family of words in F which are known as basic commutators and play an essential role. Every basic commutator u has a weight denoted by u which is a natural number. Also the basic commutators can be ordered generally with respect to their weight. Definition. Basic Commutators 1 x1 . xn are basic commutators of weight 1 and are ordered with respect to each other here x1 xn 2 if the basic commutators of weights less than n are defined then the basic commutators of weight n are w u v u 1v 1uv where i u v are basic commutators and u v n ii u v and if u s t then t v. If u n then u w. The basic commutators of weight n are ordered arbitrarily with respect to each other. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 N15 1 The following theorem of Marshall Hall plays a basic role in the study of basic commutators. Recall that the commutator subgroups 7k G in a group G are defined recursively by 71 G G and 7i 1 G t g G h x g x 2 7i G 9 2 Gi for all i 1. We refer the reader to 1 for some basic properties of 7k G . Theorem 1.1. Marshall Hall 1 Theorem 11.2.4 If F is the free group with free generators x1 . xn and if c1 . cm is the sequence of basic commutators of weights 1 . k then an arbitrary .