Sums and Rational Multiples of q-Automatic Sequences are q-Automatic
Sums and Rational Multiples of q-Automatic Sequences are q-Automatic
复制标题
q-自动序列的和和有理倍数是 q-自动
DOI：
10.1016/0304-3975(93)90202-5
复制
发表时间：
1993
期刊：
影响因子：
--
通讯作者：
Siegfried Lehr
中科院分区：
文献类型：
--
作者：
Siegfried Lehr
Christol et al.（1980）证明，对于任何素数p，序列x=（x1，x2，...，）在集合{0，1，...，p− 1}上是p-自动的当且仅当形式幂级数x（t）=∑ k= 0∞ x <$k t k在特征为p的有限域K上的函数域K（t）上是代数的，其中^是{0，1，...，p− 1}转换为K。这意味着，在形式幂级数的域K [[t]]内，这些元素x（t）的和与积再次是p-自动的。在这篇文章中，我们证明了，如果t被一个整数r的倒数代替，那么相应的真实的数集合x（r-1）的自然展开式在加有理数和乘有理数下是封闭的。
Abstract Christol et al.(1980) proved that, for any prime p, a sequence x=(x 1, x 2,…,) over the set {0, 1,…, p− 1} is p-automatic iff the formal power series x (t)=∑ k= 0∞ x ̂ k t k is algebraic over the function field K (t) over some finite field K of characteristic p, where^ is an injective mapping of {0, 1,…, p− 1} into K. This implies that, within the field K [[t]] of formal power series, sums and products of such elements x (t) are again p-automatic. In this article it is proved that, if t is replaced by the reciprocal of an integer r⩾ 2, a natural expansion of the corresponding set of real numbers x (r− 1) is closed under addition and under multiplication by rationals.