形式幂级数∏^(∞)_(n=0)(1-x^(2^(n)))^(m)系数的无界性

On the unboundedness of the coefficients of power series ∏^(∞)_(n=0)(1-x^(2^(n)))^(m)

设∏^(∞)_(n=0)(1-x^(2^(n)))^(m)为Prouhet-Thue-Morse序列的生成级数.设m≥2为正整数.令F_(m)(x)=(F(x))^(m)=(∏^(∞)_(n=0)(1-x^(2^(n)))^(m):=∑^(∞)_(n=0)t_(m)(x)x^(n).2018年,Gawron,Miska和Ulas猜想:当m≥2时序列{t_(m)(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{t_(5)(n)}^(∞)_(n=1)和{t_(6)(n)}^(∞)_(n=1)的无界子列,以及{t_(5)(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{t_(5)(n)}^(∞)_(n=1)的2-adic赋值的猜想部分成立.

Let∏^(∞)_(n=0)(1-x^(2^(n)))^(m)be the generating function of the Prouhet-Thue-Morse sequence.Let F_(m)(x)=(F(x))^(m)=(∏^(∞)_(n=0)(1-x^(2^(n)))^(m):=∑^(∞)_(n=0)t_(m)(x)x^(n).In 2018,Gawron,Miska and Ulas proposed a conjecture on the unboundedness of the sequence{t_(m)(n)}^(∞)_(n=1) for m≥2.They also proved this conjecture for m=3 and m=2k by studying the 2-adic of t_(m)(n),where k is a positive integer.In this paper,we introduce a new method for this conjecture.In this method,a class of anti-centrosymmetric matrices are firstly obtained by studying the recursive relation of tm(n).Then the conjecture may be proved by calculating the eigenvalues of the matrices.In particular,we prove the conjecture for m=5 and 6 by presenting unbounded sub-sequences of{t_(5)(n)}^(∞)_(n=1) and{t_(6)(n)}^(∞)_(n=1).Meanwhile,we also partially prove another conjecture on the 2-adic values of t_(5)(n)by calculating the 2-adic value of a sub-sequence of{t_(5)(n)}^(∞)_(n=1).