TY - JOUR

T1 - Algebraic Independence of the Values of Power Series and Their Derivatives Generated by Linear Recurrences

AU - Ide, Haruki

AU - Tanaka, Taka Aki

AU - Toyama, Kento

N1 - Funding Information:
where λil=0∑Lcli(−ki)l r0). This implies that {g0i−(z) | r0} is linearly dependent over Q modulo Q[z], and the proof is completed in a similar way to (3.10) and thereafter. □ ACKNOWLEDGMENTS. The authors are grateful to the anonymous referee for careful reading and insightful comments that improved this paper. This work was supported by JSPS KAKENHI Grant Numbers JP20J21203 and JP20K03519.
Publisher Copyright:
© 2022 International Academic Printing Co. Ltd.. All rights reserved.

PY - 2022/12

Y1 - 2022/12

N2 - Using a descent method, we construct certain power series generated by linear recurrences, each of which possesses the following property: The infinite set consisting of all its values and all the values of its derivatives of any order, at any nonzero algebraic numbers within its domain of existence, is algebraically independent. The main theorems of this paper assert that the power series of the form ∞k=0 zek, where {ek}k≥0 is a linear recurrence with certain admissible properties, have this property. In particular, Main Theorem 1.16 provides a class of {ek}k≥0 which is simpler than ever before.

AB - Using a descent method, we construct certain power series generated by linear recurrences, each of which possesses the following property: The infinite set consisting of all its values and all the values of its derivatives of any order, at any nonzero algebraic numbers within its domain of existence, is algebraically independent. The main theorems of this paper assert that the power series of the form ∞k=0 zek, where {ek}k≥0 is a linear recurrence with certain admissible properties, have this property. In particular, Main Theorem 1.16 provides a class of {ek}k≥0 which is simpler than ever before.

KW - Algebraic independence, Mahler’s method

UR - http://www.scopus.com/inward/record.url?scp=85148088574&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85148088574&partnerID=8YFLogxK

U2 - 10.3836/tjm/1502179362

DO - 10.3836/tjm/1502179362

M3 - Article

AN - SCOPUS:85148088574

SN - 0387-3870

VL - 45

SP - 519

EP - 545

JO - Tokyo Journal of Mathematics

JF - Tokyo Journal of Mathematics

IS - 2

ER -