TY - JOUR

T1 - Arithmetical properties of a certain power series

AU - Nishioka, Kumiko

AU - Shiokawa, Iekata

AU - Tamura, Jun Ichi

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 1992/9

Y1 - 1992/9

N2 - The function f(θ, φ; x, y) = Σk = 1∞ Σ1 ≤ m ≤ kθ + φ xkym, where θ > 0 is irrational and φ is real, satisfies Mahler-type functional equations which enable us to represent it by a gap-like series and then by a continued fraction. Using these representations, we describe the sequence {[(k + 1) θ + φ] - [kθ + φ]}k = 1∞ by a chain of substitutions and give algebraic independence results for the values of f(θ, φ, x, y) at some algebraic points when the partial quotients of the continued fraction of θ are unbounded, and irrationality measures for the values at some rational points.

AB - The function f(θ, φ; x, y) = Σk = 1∞ Σ1 ≤ m ≤ kθ + φ xkym, where θ > 0 is irrational and φ is real, satisfies Mahler-type functional equations which enable us to represent it by a gap-like series and then by a continued fraction. Using these representations, we describe the sequence {[(k + 1) θ + φ] - [kθ + φ]}k = 1∞ by a chain of substitutions and give algebraic independence results for the values of f(θ, φ, x, y) at some algebraic points when the partial quotients of the continued fraction of θ are unbounded, and irrationality measures for the values at some rational points.

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U2 - 10.1016/0022-314X(92)90109-3

DO - 10.1016/0022-314X(92)90109-3

M3 - Article

AN - SCOPUS:38249010672

VL - 42

SP - 61

EP - 87

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 1

ER -