Patterns of Continued Fractions for the Analogues of e and Related Numbers in the Function Field Case
نویسنده
چکیده
Patterns for simple continued fractions of the analogues of (xe 2Âf +y)Â(ze 2Âf +w) in the F q [t] case are described. In contrast to the classical case where they consist of arithmetic progressions, in this case they involve an interesting inductive scheme of block repetition and reversals, especially for q=2. 1997 Academic Press This is a sequel to [T2]. For perspective, motivation, and background material on continued fractions and the Carlitz exponential, the reader may profit from looking at [BS, T1, T2], but this paper is mostly self-contained. We start with a brief introduction: 0.0. Classical results of Euler and Hurwitz ([H1], [P]) show that the sequence of partial quotients for the (simple) continued fraction for (xe 2Âf +y)Â(ze 2Âf +w), with x, y, z, w, f integers, f {0, xw&yz{0 (the so-called Hurwitz numbers), eventually consist of a fixed number of arithmetic progressions. Examples. There is nò`formula'' giving the sequence of partial quotients: the reduction process giving a procedure to work out the sequence is quite involved. See [P, R, MW, vP] for more information.
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