Generalized k-regular sequences III: Arithmetical properties of generalized k-regular series · arXivDesk
1807.09070Jul 24, 201812pages. Major revision and Title change. We revised this paper as the series of "Generalized k-regular sequences I: Fundamental properties and examples"arXiv:1809.09320v2. In the future,we may be put together the series into one single paper
Generalized k-regular sequences III: Arithmetical properties of generalized k-regular series
Let F(z) be a k-regular series in Z[[z]] and b be an integer with b≥2
Nearby in the stack
. Bell, Bugeaud and Coons [BelBC] proved that
F(b1)
is either rational or transcendental. In [Mi], we introduce a generalized
k
-regular sequence as a unification of several kinds of important sequences including
k
-regular,
k
-additive and
k
-multiplicative sequences. In this paper, we give a generalization of the result of Bell, Bugeaud and Coons for certain generalized
k
-regular series. Especially, we show that the values of irrational generating functions of certain sum of
k
-additive sequences and certain
k
-multiplicative sequences are either rational or transcendental. Moreover, we also give a partly generalization of a result obtained by Tachiya[Ta]. Especially, we show that the values of irrational generating functions of certain
k
-additive sequences and certain
k
-multiplicative sequences give transcendental numbers.