形式幂级数∏∞n=0(1-x2n)m系数的无界性
On the unboundedness of the coefficients of power series ∏∞n=0(1-x2n)m
朱朝熹 1赵伟1
作者信息
- 1. 保密通信全国重点实验室, 成都 610041
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摘要
设∏∞n=0(1-x2n)为Prouhet-Thue-Morse序列的生成级数.设m≥2为正整数.令Fm(x)=(F(x))m=(∏∞n=0(1-x2n))m≔∑∞n=0tm(n)xn.2018年,Gawron,Miska和 Ulas猜想:当m≥2 时序列{tm(n)}∞n=1无界.对于m=3 及m=2k的情形,他们通过研究tm(n)的2-adic赋值证明猜想部分成立.本文发展了一种新方法,即由序列{tm(n)}∞n=1的递推关系式得到一类反中心对称矩阵,然后通过计算其相应矩阵的特征值来证明猜想.利用这种方法,本文证明当m=5和6时猜想成立.此外,本文还给出了序列{t5(n)}∞n=1和{t6(n)}∞n=1的无界子列,以及{t5(n)}∞n=1的一个子列的2-adic 赋值表达式,进而证明了另一个关于{t5(n)}∞n=1的2-adic赋值的猜想部分成立.
Abstract
Let ∏∞n=0(1-x2n)be the generating function of the Prouhet-Thue-Morse sequence.Let Fm(x)=(F(x))m=(∏∞n=0(1-x2n))m≔∑∞n=0tm(n)xn.In 2018,Gawron,Miska and Ulas proposed a conjecture on the unboundedness of the sequence{tm(n)}∞n=1 for m≥2.They also proved this conjecture for m=3 and m=2k by studying the 2-adic of tm(n),where k is a positive integer.In this paper,we intro-duce a new method for this conjecture.In this method,a class of anti-centrosymmetric matrices are firstly ob-tained by studying the recursive relation of tm(n).Then the conjecture may be proved by calculating the ei-genvalues of the matrices.In particular,we prove the conjecture for m=5 and 6 by presenting unbounded sub-sequences of{t5(n)}∞n=1 and{t6(n)}∞n=1.Meanwhile,we also partially prove another conjecture on the 2-adic values of t5(n)by calculating the 2-adic value of a sub-sequence of{t5(n)}∞n=1.
关键词
Prouhet-Thue-Morse/序列/无界性/特征值Key words
Prouhet-Thue-Morse sequence/Unboundedness/Eigenvalue引用本文复制引用
基金项目
保密通信重点实验室项目(61421030111012101)
出版年
2024
国家科技期刊平台
NSTL
万方数据