A Simple Proof of a Remarkable Continued Fraction Identity

Transcription

1 A Simple Proof of a Remarkale Continued Fraction Identity P. G. Anderson, T. C. Brown and P. J.-S. Shiue Citation data: P.G. Anderson, T.C. Brown, and P.J.-S. Shiue, A simple proof of a remarkale continued fraction identity, Proc. Amer. Math. Soc. 23 (995), Astract We give a simple proof of a generalization of the equality n= 2 = [2;20 ;2 ;2 ;2 2 ;2 3 ;2 5 ; : : :]; [n=τ] where τ = ( +p 5)=2 and the exponents of the partial quotients are the Fionacci numers, and some closely related results. P. E. Böhmer [2], L. V. Danilov [4], and W. W. Adams and J. L. Davison [] showed independently that if α > 0 is irrational, > is an integer, and S (α) = ( ) k=, then the simple continued [k=α] fraction for S (α) can e descried explicitly in the following way. Let α have simple continued fraction α = a 0 + a + a 2 + = [a 0 ;a ; : : :]; with p n = [a 0 ; : : : ;a n ], n 0. Let t 0 = a 0, t n = qn qn 2 qn, n. Then S (α) = [t 0 ;t ; : : :]. Thus in the case α = τ = ( + p 5)=2, the golden ratio, and = 2, one gets the remarkale equality n= 2 [n=τ] = [2;2 0 ;2 ;2 2 ;2 3 ;2 5 ; : : :], where the exponents of the partial quotients are the Fionacci numers. More recently, R. L. Graham, D. E. Knuth, and O. Patashnik [7] indicated how to give a very different proof of the power series version of this result, where the numer is replaced y an indeterminate (they carried out the proof for the case α = ( + p 5)=2), using the continuant polynomials of Euler [5]. In this note we give a proof, which we feel is simpler than the others, which makes use of a property of the characteristic sequence of α discovered y H. J. S. Smith [2]. The crucial idea of our approach appears in Lemma 2 elow, where we regard certain initial segments of the characteristic sequence of α as ase representations of integers. Department of Computer Science, Rochester Institute of Technology, Rochester, New York pga@cs.rit.edu Department of Mathematics and Statistics, Simon Fraser University, Burnay, British Columia, V5A S6, Canada. trown@sfu.ca Department of Mathematical Sciences, University of Nevada, Las Vegas, NV, USA shiue@nevada.edu

2 (Böhmer, Danilov, and Adams and Davison also show that S (α) is transcendental for every irrational α. We omit the proof of this fact, which is an easy application of a theorem of Roth [0], using Lemma 3 and Theorem B elow.) Preliminaries. Let α e an irrational numer with 0 < α <. (At the end, we will remove the restriction α <.) Let α = [0;a ;a 2 ; : : :] and p n = [0;a ; : : : ;a n ], n 0, where p n, are relatively prime non-negative integers. (As usual, we put p 2 = 0, p =, q 2 =, q = 0, so that p n = a n p n + p n 2, = a n + 2 for all n 0.) For n, define f α (n) = [(n + )α] [nα], and consider the infinite inary sequence f α = ( f α (n)) n, which is sometimes called the characteristic sequence of α. Define inary words X n, n 0, y X 0 = 0, X = 0 a, X k = X a k k X k 2, k 2, where X a denotes the word X repeated a times, and X = if a =. The following result was first proved y Smith [2]. Other proofs can e found in [3, 6,, 3], and further references to the characteristic sequence can e found in [3]. Nishioka, Shiokawa, and Tamura [8] treat the more general case [(n + )α + β ] [nα + β ]. Lemma. For each n, X n is a prefix of f α. That is, X n = f α () f α (2) f α (s), where s is the length of X n. The main proof. We are now ready to prove the result stated in the Introduction. (However, we will keep the restriction α < until the following section.) Let > e an integer, let 0 < α < e irrational, α = [0;a ;a 2 ; : : :], let p n = [0;a ; : : : ;a n ], n 0, and let the inary words X n, n 0, e defined as aove. According to Lemma, the inary word X n (which has length y a trivial induction using = a n + 2 ) is identical with the inary word f α () f α (2) f α ( ). If we let x n denote the integer whose ase representation is X n, i.e., x n = f α () + fα (2) fα ( ) 0, then we can write Now we come to the crucial step. x n = k= k : Lemma 2. For n 0, let t n+ = + n. Then for n, x n+ = t n+ x n + x n : Proof. Using the facts that X n has length, X n has length, x n+ is the integer whose ase representation is X n+, and X n+ = X a n+ n X n, it follows that x n+ = ( + qn + 2qn + + (a n+ ) )x n + x n = ( a n+ ) ( q x n n + x n = t n+ x n + x n ) Lemma 3. For n, [0;t ; : : : ;t n ] = x n: 2

3 Proof. Let y n = qn, n 0. We show y induction on n that [0;t ; : : : ;t n ] = x n y n. We start the induction at n = 0 y setting t 0 = 0. Note that x 0 = 0, x =, y 0 =, y = q = t. For the induction step, we simply note that x n+ = t n+ x n + x n and y n+ = t n+ y n + y n. Theorem A. Let > e an integer, and let 0 < α < e irrational, with f α (n) = [(n + )α] [nα], n. Let α = [0;a ;a 2 ; : : :], let p n q = n [0;a ; : : : ;a n ], n 0 (where p n ; are relatively prime non-negative integers), and let t n = qn qn 2, n. Then qn ( ) k= k = [0;t ;t 2 ; : : :]: Proof. We have seen that x n = k=. Hence y Lemma 3, k ( ) and we can take the limit as n!. qn k= k = [0;t ; : : : ;t n ]; Theorem B. With the same hypotheses as in Theorem A, we have ( ) n= = [0;t ;t 2 ; : : :]: [n=α] Proof. This is a restatement of Theorem A, using the easily verified fact (when 0 < α < ) that = if and only if k = [n=α] for some n. Theorem C. With the same hypotheses as in Theorem A, we have ( ) 2 [kα] k= k = [0;t ;t 2 ; : : :]: Proof. Using = [(k + )α] [kα] and [α] = 0, the series in Theorem C is otained from the series in Theorem A y a slight rearrangement. Theorem D. With the same hypotheses as in Theorem A, we have k= k = ( ) k= ( ) k ( q k )( q k ) : Proof. We say in the proof of Lemma 3 that [0;t ; : : : ;t n ] = x n y n, n, where y n = qn, n 0. By a well-known theorem (J. B. Roerts [9, pp. 0]), x n y = n n ( ) k k= y k y, n, and Theorem D now follows k from Theorem A. Removing the restriction α <. 0 < α <. Now let α 0 = a 0 + α, where a 0 0 is an integer, α is irrational, and 3

4 By Theorem A we get f ( ) α 0 (k) k= k = ( ) k= = ( )a 0 k= a 0 + k = a 0 + [0;t ;t 2 ; : : :] = [a 0 ;t ;t 2 ; : : :]: + ( ) k k= k To handle Theorem B we need to use the fact, whose simple proof we omit, that if α 0 = a 0 + α, where 0 < α <, then for each k = 0;;2; : : :, the value k is assumed y the expression [n=α 0 ] exactly a 0 + times if [n=α] = k for some n, and exactly a 0 times if [n=α] never equals k. It then follows from Theorem B that ( ) n= [n=α0 ] = [a 0;t ;t 2 ; : : :]. By Theorem C and some careful rearrangement we get ( ) 2 k= Finally, the modified Theorem D (using the modified Theorem A) is f ( ) α 0 (k) k= k = a 0 + k= ( ) k ( ) 2 ( q k )( q k ) : [kα 0 ] k = [a 0 ;t ;t 2 ; : : :]. Remark. This paper grew out of the first author s consideration of the numer k=, where α 2 k = +p 5 2, as the fixed point of the sequence fg n (0)g, n, where g (x) = x=2, g 2 (x) = (x + )=2, g n (x) = g n (g n 2 (x)), n 3. This quickly leads (upon setting g n (x) = (x + a n )= n and solving for a n and n ) to k= 2 k = [2;2 0 ;2 ;2 ;2 2 ;2 3 ;2 5 ; : : :]: Acknowledgement. The authors are grateful to the referee for references [2] and [4] and for several helpful remarks. References [] W.W. Adams and J.L. Davison, A remarkale class of continued fractions, Proc. Amer. Math. Soc. 65 (977), [2] P.E. Böhmer, Üer die transzendenz gewisser dyadischer rüche, Math. Ann. 96 (926), , erratum 96 (926) 735. [3] T.C. Brown, Descriptions of the characteristic sequence of an irrational, Canad. Math. Bull. 36 (993), 5 2. [4] L.V. Danilov, Some classes of transcendental numers, Math. Notes Acad. Sci. USSR 2 (972), [5] L. Euler, Specimen algorithmi singularis, Novi Commentarii Academiae Cientiarum Petropolitanae 9 (762), 53 69, Reprinted in his Opera Omnia, Series, Vol. 5, pp

5 [6] A.S. Fraenkel, A. Mushkin, and U. Tassa, Determination of [nθ ] y its sequence of differences, Canad. Math. Bull. 2 (978), [7] R.L. Graham, D.E. Knuth, and O. Patashnik, Concrete mathematics, Addison-Wesley, New York, 989. [8] K. Nishioka, I. Shiokawa, and J. Tamura, Arithmetical properties of a certain power series, J. Numer Theory 42 (996), [9] J.B. Roerts, Elementary numer theory, MIT Press, Boston, 977. [0] Klaus F. Roth, Rational approximations to algeraic numers, Mathematika 2 (955), 20; corrigendum 2 (955), 68. [] Jeffrey Shallit, Characteristic words as fixed points of homomorphisms, Tech. Report CS-9-72, Univ. of Waterloo, Dep. of Computer Science, 99. [2] H.J.S. Smith, Note on continued fractions, Messenger Math. (6) (876), 4. [3] K.B. Stolarsky, Beatty sequences, continued fractions, and certain shift operators, Canad. Math. Bull. 9 (976),