Two types of explicit continued fractions
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TWO TYPES OF EXPLICIT CONTINUED FRACTIONS Pakwan Riyapan1 (Pattani), Vichian Laohakosol2 (Bangkok) and Tuangrat Chaichana3 (Bangkok) [Communicated by: Attila Peth˝ o] 1
Department of Mathematics and Computer Science Faculty of Science and Technology, Prince of Songkla University Pattani Campus 94000, Thailand E-mail: [email protected] 2
3
Department of Mathematics, Faculty of Science, Kasetsart University Bangkok 10900, Thailand, E-mail: [email protected]
Department of Mathematics, Faculty of Science, Chulalongkorn University Bangkok 10330, Thailand, E-mail: t [email protected] (Received: November 10, 2005; Accepted: January 13, 2006)
Abstract Two types of explicit continued fractions are presented. The continued fractions of the first type include those discovered by Shallit in 1979 and 1982, which were later generalized by Peth˝ o. They are further extended here using Peth˝ o’s method. The continued fractions of the second type include those whose partial denominators form an arithmetic progression as expounded by Lehmer in 1973. We give here another derivation based on a modification of Komatsu’s method and derive its generalization. Similar results are also established for continued fractions in the field of formal series over a finite base field.
1. Introduction In recent years, there have been numerous results involving explicit series expansions of certain continued fractions. Consider, for example, the following results due to Shallit ([6], [7]). I. (Shallit [6]) Let v 1 , B(u, v) = 2k u k=0 Mathematics subject classification number: 11A55, 11F70. Key words and phrases: explicit continued fractions, Peth˝ o’s method, Komatsu’s method. 0031-5303/2006/$20.00 c Akad´ emiai Kiad´ o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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p. riyapan, v. laohakosol and t. chaichana
where u is an integer ≥ 3. (A) We have B(u, 0) = [0; u], B(u, 1) = [0; u − 1, u + 1]. (B) If B(u, v) = [a0 ; a1 , . . . , an ], then B(u, v + 1) = [a0 ; a1 , . . . , an−1 , an + 1, an − 1, an−1 , an−2 , . . . , a2 , a1 ]. ∞
II. (Shallit [7]) Let {c(k)}k=0 be a sequence of positive integers such that c(v + 1) ≥ 2c(v) for all v ≥ v , where v ∈ N0 . Let d(v) := c(v + 1) − 2c(v) and define S(u, v) =
v k=0
1 . uc(k)
For v ≥ v , if S(u, v) = [a0 ; a1 , . . . , an ] and n is even, then S(u, v + 1) = [a0 ; a1 , . . . , an , ud(v) − 1, 1, an − 1, an−1 , an−2 , . . . , a2 , a1 ]. These two results of Shallit were later generalized by Peth˝ o ([5]). III. (Peth˝ o [5]) Let {ai }, with a1 ≥ 2 be an infinite sequence of positive integers, and d1 = 1, di = ±1 for i ≥ 2. Let Qi be defined by the recursion Q1 = 1, Qi = ai−1 Qki−1 for i ≥ 2, where k ≥ 2 is an integer. Finally let Ck (a, u) =
u di . Qi i=1
Then Ck (a, 1) = [1], and Ck (a, 2) = [1; a1 − 1, 1], = [0; 1, a1 − 1],
if d2 = 1, if d2 = −1.
Let Ck (a, u) = [A0 ; A1 , . . . , An ] for u ≥ 2, and its length be chosen odd or even according as du+1 = −1 or du+1 = 1. Then Ck (a, u + 1) − 1, An−1 + 1, An−2 , . . . , A1 ], = [A0 ; A1 , . . . , An , au Qk−2
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