### Abstract

The function f(θ, φ; x, y) = Σ_{k = 1}^{∞} Σ_{1 ≤ m ≤ kθ + φ} x^{k}y^{m}, 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.

Original language | English |
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Pages (from-to) | 61-87 |

Number of pages | 27 |

Journal | Journal of Number Theory |

Volume | 42 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1992 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

*42*(1), 61-87. https://doi.org/10.1016/0022-314X(92)90109-3

**Arithmetical properties of a certain power series.** / Nishioka, Kumiko; Shiokawa, Iekata; Tamura, Jun Ichi.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 42, no. 1, pp. 61-87. https://doi.org/10.1016/0022-314X(92)90109-3

}

TY - JOUR

T1 - Arithmetical properties of a certain power series

AU - Nishioka, Kumiko

AU - Shiokawa, Iekata

AU - Tamura, Jun Ichi

PY - 1992

Y1 - 1992

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.

UR - http://www.scopus.com/inward/record.url?scp=38249010672&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38249010672&partnerID=8YFLogxK

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 -