A simple proof of a remarkable continued fraction identity
Authors:
P. G. Anderson, T. C. Brown and P. J.S. Shiue
Journal:
Proc. Amer. Math. Soc. 123 (1995), 20052009
MSC:
Primary 11A55
MathSciNet review:
1249866
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We give a simple proof of a generalization of the equality where and the exponents of the partial quotients are the Fibonacci numbers, and some closely related results.
 [1]
William
W. Adams and J.
L. Davison, A remarkable class of continued
fractions, Proc. Amer. Math. Soc.
65 (1977), no. 2,
194–198. MR 0441879
(56 #270), http://dx.doi.org/10.1090/S00029939197704418794
 [2]
Tom
C. Brown, Descriptions of the characteristic sequence of an
irrational, Canad. Math. Bull. 36 (1993), no. 1,
15–21. MR
1205889 (94g:11051), http://dx.doi.org/10.4153/CMB19930036
 [3]
P. E. Böhmer, Über die Transzendenz gewisser dyadischer Brüche, Math. Ann. 96 (1926), 367377; erratum 96 (1926), 735.
 [4]
L.
V. Danilov, Certain classes of transcendental numbers, Mat.
Zametki 12 (1972), 149–154 (Russian). MR 0316391
(47 #4938)
 [5]
J.
L. Davison, A series and its associated continued
fraction, Proc. Amer. Math. Soc.
63 (1977), no. 1,
29–32. MR
0429778 (55 #2788), http://dx.doi.org/10.1090/S00029939197704297785
 [6]
L. Euler, Specimen algorithmi singularis, Novi Commentarii Academiae Cientiarum Petropolitanae 9 (1762), 5369; reprinted in his Opera Omnia, Series 1, Vol. 15, pp. 3149.
 [7]
A.
S. Fraenkel, M.
Mushkin, and U.
Tassa, Determination of [𝑛𝜃] by its sequence of
differences, Canad. Math. Bull. 21 (1978),
no. 4, 441–446. MR 523586
(80d:10051), http://dx.doi.org/10.4153/CMB19780770
 [8]
Ronald
L. Graham, Donald
E. Knuth, and Oren
Patashnik, Concrete mathematics, 2nd ed., AddisonWesley
Publishing Company, Reading, MA, 1994. A foundation for computer science.
MR
1397498 (97d:68003)
 [9]
Kumiko
Nishioka, Iekata
Shiokawa, and Junichi
Tamura, Arithmetical properties of a certain power series, J.
Number Theory 42 (1992), no. 1, 61–87. MR 1176421
(93i:11086), http://dx.doi.org/10.1016/0022314X(92)901093
 [10]
J. B. Roberts, Elementary number theory, MIT Press, Boston, 1977.
 [11]
K.
F. Roth, Rational approximations to algebraic numbers,
Mathematika 2 (1955), 1–20; corrigendum, 168. MR 0072182
(17,242d)
 [12]
J. Shallit, Characteristic words as fixed points of homomorphisms, Univ. of Waterloo, Dept. of Computer Science, Tech. Report CS9172, 1991.
 [13]
H. J. S. Smith, Note on continued fractions, Messenger Math. 6 (1876), 114.
 [14]
Kenneth
B. Stolarsky, Beatty sequences, continued fractions, and certain
shift operators, Canad. Math. Bull. 19 (1976),
no. 4, 473–482. MR 0444558
(56 #2908)
 [1]
 W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194198. MR 0441879 (56:270)
 [2]
 T. C. Brown, Descriptions of the characteristic sequence of an irrational, Canad. Math. Bull. 36 (1993), 1521. MR 1205889 (94g:11051)
 [3]
 P. E. Böhmer, Über die Transzendenz gewisser dyadischer Brüche, Math. Ann. 96 (1926), 367377; erratum 96 (1926), 735.
 [4]
 L. V. Danilov, Some classes of transcendental numbers, Math. Notes Acad. Sci. USSR 12 (1972), 524527. MR 0316391 (47:4938)
 [5]
 J. L. Davidson, A series and its associated continued fraction, Proc. Amer. Math. Soc. 63 (1977), 2932. MR 0429778 (55:2788)
 [6]
 L. Euler, Specimen algorithmi singularis, Novi Commentarii Academiae Cientiarum Petropolitanae 9 (1762), 5369; reprinted in his Opera Omnia, Series 1, Vol. 15, pp. 3149.
 [7]
 A. S. Fraenkel, M. Mushkin, and U. Tassa, Determination of by its sequence of differences, Canad. Math. Bull. 21 (1978), 441446. MR 523586 (80d:10051)
 [8]
 R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete mathematics, AddisonWesley, New York, 1989. MR 1397498 (97d:68003)
 [9]
 K. Nishioka, I. Shiokawa, and J. Tamura, Arithmetical properties of a certain power series, J. Number Theory 42 (1992), 6187. MR 1176421 (93i:11086)
 [10]
 J. B. Roberts, Elementary number theory, MIT Press, Boston, 1977.
 [11]
 K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 120; corrigendum 2 (1955), 168. MR 0072182 (17:242d)
 [12]
 J. Shallit, Characteristic words as fixed points of homomorphisms, Univ. of Waterloo, Dept. of Computer Science, Tech. Report CS9172, 1991.
 [13]
 H. J. S. Smith, Note on continued fractions, Messenger Math. 6 (1876), 114.
 [14]
 K. B. Stolarsky, Beatty sequences, continued fractions, and certain shift operators, Canad. Math. Bull. 19 (1976), 473482. MR 0444558 (56:2908)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
11A55
Retrieve articles in all journals
with MSC:
11A55
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199512498664
PII:
S 00029939(1995)12498664
Article copyright:
© Copyright 1995
American Mathematical Society
