### Abstract

The study of irrationality properties of values of the generalized Tschakaloff series f(x) defined by (1.2) below was initiated by Duverney (Portugal. Math. 53(2) (1996) 229; Period. Math. Hungar. 35 (1997) 149), and continued by the authors (J. Number Theory 77 (1999) 155). The present paper proceeds to extend our previous result (Amou and Katsurada, 1999, Theorem). The irrationality of f(α) for any α ∈ Q\{0} is proved in a quantitative form under fairly general growth conditions on the coefficients of f(x) (Theorem 1), while the same result is shown in a certain 'limiting' situation of Theorem 1, at the cost of loosing a quantitative aspect (Theorem 2). The linear independence of certain values of a system of f(x) is also obtained (Theorem 3). The key idea in proving our previous result is a Mahler's transcendence method, due to Loxton and van der Poorten (in: A. Baker, D.W. Masser (Eds.), Transcendence Theory: Advances and Applications, Academic Press, San Diego, 1977, pp. 211-226), applied to an appropriate sequence of functions (see (2.4) and (2.5)). In order to establish Theorems 1 and 2, this method is enhanced by a certain technique which allows us to improve zero estimates for the remainder terms of Padé-type approximations (see Lemmas 3 and 4).

Original language | English |
---|---|

Pages (from-to) | 132-155 |

Number of pages | 24 |

Journal | Journal of Number Theory |

Volume | 104 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2004 Jan |

### Fingerprint

### Keywords

- Irrationality
- Irrationality measure
- Padé approximation
- q-difference equation
- Siegel's lemma

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Irrationality results for values of generalized Tschakaloff series II.** / Amou, Masaaki; Katsurada, Masanori.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 104, no. 1, pp. 132-155. https://doi.org/10.1016/S0022-314X(03)00143-4

}

TY - JOUR

T1 - Irrationality results for values of generalized Tschakaloff series II

AU - Amou, Masaaki

AU - Katsurada, Masanori

PY - 2004/1

Y1 - 2004/1

N2 - The study of irrationality properties of values of the generalized Tschakaloff series f(x) defined by (1.2) below was initiated by Duverney (Portugal. Math. 53(2) (1996) 229; Period. Math. Hungar. 35 (1997) 149), and continued by the authors (J. Number Theory 77 (1999) 155). The present paper proceeds to extend our previous result (Amou and Katsurada, 1999, Theorem). The irrationality of f(α) for any α ∈ Q\{0} is proved in a quantitative form under fairly general growth conditions on the coefficients of f(x) (Theorem 1), while the same result is shown in a certain 'limiting' situation of Theorem 1, at the cost of loosing a quantitative aspect (Theorem 2). The linear independence of certain values of a system of f(x) is also obtained (Theorem 3). The key idea in proving our previous result is a Mahler's transcendence method, due to Loxton and van der Poorten (in: A. Baker, D.W. Masser (Eds.), Transcendence Theory: Advances and Applications, Academic Press, San Diego, 1977, pp. 211-226), applied to an appropriate sequence of functions (see (2.4) and (2.5)). In order to establish Theorems 1 and 2, this method is enhanced by a certain technique which allows us to improve zero estimates for the remainder terms of Padé-type approximations (see Lemmas 3 and 4).

AB - The study of irrationality properties of values of the generalized Tschakaloff series f(x) defined by (1.2) below was initiated by Duverney (Portugal. Math. 53(2) (1996) 229; Period. Math. Hungar. 35 (1997) 149), and continued by the authors (J. Number Theory 77 (1999) 155). The present paper proceeds to extend our previous result (Amou and Katsurada, 1999, Theorem). The irrationality of f(α) for any α ∈ Q\{0} is proved in a quantitative form under fairly general growth conditions on the coefficients of f(x) (Theorem 1), while the same result is shown in a certain 'limiting' situation of Theorem 1, at the cost of loosing a quantitative aspect (Theorem 2). The linear independence of certain values of a system of f(x) is also obtained (Theorem 3). The key idea in proving our previous result is a Mahler's transcendence method, due to Loxton and van der Poorten (in: A. Baker, D.W. Masser (Eds.), Transcendence Theory: Advances and Applications, Academic Press, San Diego, 1977, pp. 211-226), applied to an appropriate sequence of functions (see (2.4) and (2.5)). In order to establish Theorems 1 and 2, this method is enhanced by a certain technique which allows us to improve zero estimates for the remainder terms of Padé-type approximations (see Lemmas 3 and 4).

KW - Irrationality

KW - Irrationality measure

KW - Padé approximation

KW - q-difference equation

KW - Siegel's lemma

UR - http://www.scopus.com/inward/record.url?scp=0347662377&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0347662377&partnerID=8YFLogxK

U2 - 10.1016/S0022-314X(03)00143-4

DO - 10.1016/S0022-314X(03)00143-4

M3 - Article

VL - 104

SP - 132

EP - 155

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 1

ER -