The Fibonacci Word’s Hankel Determinant And Padé Approximation

This thesis studies the quantitative relation between the Hankel determinants and thePadé approximation, and summarize some new properties of Fibonacci word. For words,we introduce a general theory of Padé approximation, and for the fixed point ξ of a sub-stitution θ defined by θ(a)=ab, θ(b)=a, give explicit formulae of Padé approximantspn/qn. It is hard to pin down precisely who first introduced the Fibonacci words. The infi-nite Fibonacci word is a simple of a sturmian word. In half a century, people found manyimportant and interesting properties of this word again and again from different fields andaspects. In the paper of Knuth, Morris, Pratt[1977] Fibonacci strings appear as the worstcase of a pattern-matching algorithm, Crochemore[1981] used Fibonacci word to show theoptimality of an algorithm for computing square subwords. The combinatorial properties ofthe infinite Fibonacci word are of great interest in some aspects of mathematics and physics,such as formal language, fractal geometry, number theory, computational complexity, qua-sicrystals etc.This thesis studies the quantitative relation between the Hankel determinants and thePadé approximation, and summarize some new properties of Fibonacci word. For words,we introduce a general theory of Padé approximation, and for the fixed point ξ of a substi-tution θ defined by θ(a)=ab, θ(b)=a, give explicit formulae of Padé approximants pn/qnand Hankel determinants. From these properties, I improve the proof of former conclusion,and estimate the irrationality exponent of Fibonacci word.In the preparation section, we introduce the definitions of group、ring、field、Hankel determinant Padé approximation and some lemmas related with determinantoperation. in the third chapter, namely, the core of this paper, I summarize some newproperties of Fibonacci word, computational methods of Hankel determinant of this word,the quantitative relation between the Hankel determinants and the Padé approximation.Finally, I estimate the irrationality exponent of Fibonacci word.