TY - JOUR
T1 - A criterion of algebraic independence of values of modular functions and an application to infinite products involving Fibonacci and Lucas numbers
AU - Duverney, Daniel
AU - Elsner, Carsten
AU - Kaneko, Masanobu
AU - Tachiya, Yohei
N1 - Funding Information:
The authors would like to express their gratitude to the referee for his/her careful reading of our manuscript. This work was partly supported by JSPS KAKENHI Grant Numbers JP18K03201 (Y. Tachiya) and JP16H06336, JP18K18712, JP21H04430, JP21K18141 (M. Kaneko).
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2022/6
Y1 - 2022/6
N2 - The aim of this paper is to give a criterion of algebraic independence for the values at the same point of two modular functions under certain conditions. As an application, we show that any two infinite products in ∏n=1∞(1+1Fn),∏n=3∞(1-1Fn),∏n=1∞(1+1Ln),∏n=2∞(1-1Ln)are algebraically independent over Q, where { Fn} and { Ln} are the Fibonacci and Lucas sequences, respectively. The proof of our main theorem is based on the properties of the field of all modular functions for the principal congruence subgroup, together with a deep result of Yu. V. Nesterenko on algebraic independence of the values of the Eisenstein series.
AB - The aim of this paper is to give a criterion of algebraic independence for the values at the same point of two modular functions under certain conditions. As an application, we show that any two infinite products in ∏n=1∞(1+1Fn),∏n=3∞(1-1Fn),∏n=1∞(1+1Ln),∏n=2∞(1-1Ln)are algebraically independent over Q, where { Fn} and { Ln} are the Fibonacci and Lucas sequences, respectively. The proof of our main theorem is based on the properties of the field of all modular functions for the principal congruence subgroup, together with a deep result of Yu. V. Nesterenko on algebraic independence of the values of the Eisenstein series.
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U2 - 10.1007/s40993-022-00328-7
DO - 10.1007/s40993-022-00328-7
M3 - Article
AN - SCOPUS:85129737283
SN - 2363-9555
VL - 8
JO - Research in Number Theory
JF - Research in Number Theory
IS - 2
M1 - 31
ER -