# BOUNDED PARTIAL QUOTIENTS OF SOME CUBIC POWER SERIES WITH BINARY COEFFICIENTS

• Ayadi, Khalil (Department of Mathematics Faculty of Sciences University of Sfax) ;
• Beldi, Salah (Department of Mathematics Faculty of Sciences University of Sfax) ;
• Lee, Kwankyu (Department of Mathematics Education Chosun University)
• Received : 2015.05.27
• Published : 2016.07.31

#### Abstract

It is a surprising but now well-known fact that there exist algebraic power series of degree higher than two with partial quotients of bounded degrees in their continued fraction expansions, while there is no single algebraic real number known with bounded partial quotients. However, it seems that these special algebraic power series are quite rare and it is hard to determine their continued fraction expansions explicitly. To the short list of known examples, we add a new family of cubic power series with bounded partial quotients.

#### Acknowledgement

Supported by : Chosun University

#### References

1. A. Lasjaunias, Continued fractions for algebraic formal power series over a finite base field, Finite Fields Appl. 5 (1999), no. 1, 46-56. https://doi.org/10.1006/ffta.1998.0236
2. L. E. Baum and M. M. Sweet, Continued fractions of algebraic power series in characteristic 2, Ann. of Math. (2) 103 (1976), no. 3, 593-610. https://doi.org/10.2307/1970953
3. B. de Mathan, Approximation exponents for algebraic functions in positive characteristic, Acta Arith. 60 (1992), no. 4, 359-370. https://doi.org/10.4064/aa-60-4-359-370
4. D. Gomez-Perez and A. Lasjaunias, Hyperquadratic power series in $F_{3}((T^{-1}))$ with partial quotients of degree 1, Ramanujan J. 33 (2014), no. 2, 219-226. https://doi.org/10.1007/s11139-013-9545-4
5. A. I. Khinchin, Continued Fractions, University of Chicago Press, 1964.
6. A. Lasjaunias, Quartic power series in $F_{3}((T^{-1}))$ with bounded partial quotients, Acta Arith. 95 (2000), no. 1, 49-59. https://doi.org/10.4064/aa-95-1-49-59
7. A. Lasjaunias and J.-J. Ruch, Algebraic and badly approximable power series over a finite field, Finite Fields Appl. 8 (2002), no. 1, 91-107. https://doi.org/10.1006/ffta.2000.0329
8. A. Lasjaunias and J.-J. Ruch, Flat power series over a finite field, J. Number Theory 95 (2002), no. 2, 268-288. https://doi.org/10.1016/S0022-314X(01)92764-7
9. A. Lasjaunias and J.-Y. Yao, Hyperquadratic continued fractions in odd characteristic with partial quotients of degree one, J. Number Theory 149 (2015), 259-284. https://doi.org/10.1016/j.jnt.2014.10.012
10. K. Lee, Continued fractions for linear fractional transformations of power series, Finite Fields Appl. 11 (2005), no. 1, 45-55. https://doi.org/10.1016/j.ffa.2004.04.002
11. W. H. Mills and D. P. Robbins, Continued fractions for certain algebraic power series, J. Number Theory 23 (1986), no. 3, 388-404. https://doi.org/10.1016/0022-314X(86)90083-1
12. M. Mkaouar, Sur les fractions continues des series formelles quadratiques sur $f_q(x)$, Acta Arith. 97 (2001), no. 3, 241-251. https://doi.org/10.4064/aa97-3-4
13. W. M. Schmidt, On continued fractions and diophantine approximation in power series fields, Acta Arith. 95 (2000), no. 2, 139-166. https://doi.org/10.4064/aa-95-2-139-166
14. D. S. Thakur, Diophantine approximation exponents and continued fractions for algebraic power series, J. Number Theory 79 (1999), no. 2, 284-291. https://doi.org/10.1006/jnth.1999.2413
15. J. F. Voloch, Diophantine approximation in positive characteristic, Period. Math. Hungar. 19 (1988), no. 3, 217-225. https://doi.org/10.1007/BF01850290