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:
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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
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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 -