Series and product expansions for elements in function fields and characterizations of rational elements

Abstract

It is well-known that a positive real number can be uniquely represented as an infinite series or as an infinite product in many different forms, such as, base representations, Sylvester series, Engel series, Luroth series, and Cantor products. Most of these representations also have counterparts in the file of p-adic numbers. Starting from 1987, A. Knopfmacher and J. Knopfmacher introduced two major general algorithms, one for product and the other for series expansions, which give unique representations for p-adic numbers. Such algorithms embrace all the above-mentioned expansions as special cases. The aims of this thesis are to carry over the Knopfmachers algorithms to the case of function fields and to investigate the possibility of characterizing rational elements via these expansions. By function fields, we refer to F[subscript q]((p(x)) an F[subscript q]((1/x)), the completions of the field of rational functions F[subscript q](x) with respect to the p(x)-adic valuation and the infinite valuation, respectively, where p(x) is an irreducible polynomial over F[subscript q] Following the processes similar to those of the Knopfmachers, in the first part, the algorithms for constructing series and product expansions for elements in the fields F[subscript q]((px))) an F[subscript q]((1/x)) are described. Detailed proofs of their convergence, uniqueness and the degrees of approximation are proved together with examples derivable from these algorithms. The second part deals with the problem of characterizing rational elements in both fields through their series and product expansions. Using the concept of digit set as expounded in the works of the Knopfmachers, it is found that all, but one, series expansions represent rational elements if and only if they are finite. In the exceptional case, the Luroth-type perhaps the hardest case, it is shown that rational elements correspond either to finite or periodic expansions. The characterization of rational elements is deemed complete for series expansions. This is in stark contrast to the p-adic case where this problem remains open in a few cases. Regarding product expansions, only one particular type, called Cantor product expansion, which is constructed from one of the series expansions, the Type 4 expansion, is completely characterized using the result from the series case. Sufficient conditions for rationality are established for the remaining cases. This again is more favorable than in the p-adic case where only some sufficient conditions are known for ll such product expansions.