TY - GEN
T1 - Remarkable algebraic independence property of certain series related to continued fractions
AU - Tanaka, Taka Aki
PY - 2008/12/1
Y1 - 2008/12/1
N2 - We prove, using Mahler's method, the following results: Theorem 1 asserts that the series Θx,a,q) are algebraically independent for any distinct triplets (x,a,q) of nonzero algebraic numbers, where Θ (x,a,q) has the property shown in Corollary 1 that Θ (a,a,q) is expressed as a continued fraction. Theorem 2 asserts, under the weaker condition than that of Theorem 1, that the values Θ(x,1,q) are algebraically independent for any distinct pairs (x,q) of nonzero algebraic numbers. Typical examples of these results are generated by Fibonacci numbers.
AB - We prove, using Mahler's method, the following results: Theorem 1 asserts that the series Θx,a,q) are algebraically independent for any distinct triplets (x,a,q) of nonzero algebraic numbers, where Θ (x,a,q) has the property shown in Corollary 1 that Θ (a,a,q) is expressed as a continued fraction. Theorem 2 asserts, under the weaker condition than that of Theorem 1, that the values Θ(x,1,q) are algebraically independent for any distinct pairs (x,q) of nonzero algebraic numbers. Typical examples of these results are generated by Fibonacci numbers.
KW - Algebraic independence
KW - Continued fractions
KW - Fibonacci numbers
KW - Mahler's method
UR - http://www.scopus.com/inward/record.url?scp=77958172298&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=77958172298&partnerID=8YFLogxK
U2 - 10.1063/1.2841905
DO - 10.1063/1.2841905
M3 - Conference contribution
AN - SCOPUS:77958172298
SN - 9780735404953
T3 - AIP Conference Proceedings
SP - 190
EP - 204
BT - Diophantine Analysis and Related Fields, DARF 2007/2008
T2 - Diophantine Analysis and Related Fields, DARF 2007/2008
Y2 - 5 March 2008 through 7 March 2008
ER -