Explicit Descriptions of Some Continued Fractions

Recently, Bergman [2] provided an explicit, nonrecursive description of the partial quotients in (1), and by implication, in (2). (This description is our Theorem 3.) The purpose of this paper is to prove Bergman's result, and to provide similar results for the continued fractions given in [3] and [4]. We start off with some terminology about "strings." By a string9 we mean a (finite or infinite) ordered sequence of symbols. Thus, for example, we may consider the partial quotients [<ZQ $ &i 9 • • • 9 CLn\ of a continued fraction to be a string. If w and x are strings, then by wx9 the concatenation of w with x9 we mean the juxtaposition of the elements of w with those of x. By \w\ 9 we mean the length of w9 i.e., the number of symbols in w. Note that |i<;| may be either 0 or oo. If w is a finite string, then by w9 the reversal of w9 we mean the symbols of w taken in reverse order. Finally, by the symbol W, we mean the string WWW ... W