The Research On K-regular Sequences

The research for automatic sequence with great theoretic and practical implications,has been widespread concern. Unfortunately, the range of automatic sequences is necessarily finite, and this restricts their descriptive power. Hence, in this paper, we generalize the concept of automatic sequence to the case where the sequence can take its values in a(possibly infinite) ring, we call such sequences k-regular.The first chapter, describes the research background and status of automatic sequences,and introduces the main content of this paper.The second chapter, provides the basic concepts of rings and fields, module theory in algebra, and the definition, some properties and equivalent characterization of automatic sequences.The third chapter, introduces the definition and some equivalent descriptions of kregular sequences. We prove that k-regular sequences and k-automatic sequences are equivalent under certain condition. We investigate some properties of regular sequences, such as,a sequence is kf-regular if and only if it is k-regular, the growth of k-regular sequences is exponential. Finally, we give some sequences, and prove that they are k-regular.The fourth chapter, describes the definition of k-regular power series, proves that kregular sequences have nice closure properties, and the set of k-regular sequences forms a ring under the operations of term-by-term addition and convolution, while the set of kregular power series also forms a ring under addition and ordinary multiplication. Finally,We define the units of the k-regular sequences, and discuss the regularity of the units in a case of 3-regular sequence.The fifth chapter, on the basics of the result of Morton and Mourant, we prove that every sequence(s(n))n?0over Z is regular if and only if its pattern transform(?s(n))n?0is regular. In particular, the coefficients in the expansion of(ep(an + b))n?0form a k-automatic sequence.