Irrationality results for values of generalized Tschakaloff series
type: Item
instance of: scholarly article
title: Irrationality results for values of generalized Tschakaloff series
zbMATH Open document ID: 0929.11020
author: Masaaki Amou
author: Masanori Katsurada
published in: Journal of Number Theory
publication date: 2000-01-31T00:00:00Z
review text: The series of the title takes the shape \[ f(x)= \sum^\infty_{n=0} {s_nu_0 u_1\dots u_n\over t_nv_0 v_1\dots v_n} {x^n\over q^{n (n+1)/2}}, \] where \(s_n,t_n, u_n,v_n\) are sequences of non-zero integers restricted in absolute value by \(\exp (cn^\delta)\) with \(0<\delta <{1\over 4}\) and \(c>0\). The main result states that if \(\alpha\) is a non-zero real number and \(q\) is an integer with \(| q|\geq 2\), then \(f(\alpha)\) is irrational and has a measure of irrationality satisfying \[ \left| f(\alpha) -{M\over N}\right |> \exp \bigl\{-c_1 (\log N)^{8/5} \bigr\}, \] where \(c_1\) is a positive constant depending only on \(f\) and \(\alpha\). The result extends and improves a number of results on Tschakaloff's series (the case with \(s_n=t_n =u_n= v_n=1)\) and its extensions. The proof follows closely Mahler's method in transcendental number theory. It exploits the functional equations connecting \(f(qx)\) and \(f(x)\) and is an attractive new application of the method.
Mathematics Subject Classification ID: 11J82
zbMATH DE Number: 1309273
zbMATH Keywords: irrationality
zbMATH Keywords: Chakalov series
zbMATH Keywords: measure of irrationality
zbMATH Keywords: Tschakaloff's series
zbMATH Keywords: Mahler's method
zbMATH Keywords: functional equations
reviewed by: John H. Loxton
MaRDI profile type: Publication
full work available at URL: jnth.1999.2376
OpenAlex ID: W4212830119
cites work: Q3960772
cites work: Linear independence of the values of transcendental solutions of certain functional equations.
cites work: Indépendance linéaire des valeurs des solutions transcendantes de certaines équations fonctionnelles II
cites work: Q4402396
cites work: Linear independence measures for certain numbers
cites work: Q4886387
cites work: On arithmetic properties of Tschakaloff's function
cites work: Über die lineare Unabhängigkeit von Werten einer speziellen Reihe. (On the linear independence of values of a special series)
cites work: Q3257865
cites work: Linear independence measures for values of Heine series
cites work: Q4156490
cites work: Q5814004
cites work: Arithmetic properties of special Heine series
cites work: Zu einem Satz von Skolem über lineare Unabhängigkeiten von Werten gewisser Thetareihen. (On a theorem of Skolem on linear independence of values of certain theta series)
cites work: A linear independence measure for certain \(p\)-adic numbers
DOI: 10.1006/JNTH.1999.2376
description: scientific article; zbMATH DE number 1309273