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 -