On the regularity of the Hankel determinant sequence of the characteristic sequence of powers of 2

On the regularity of the Hankel determinant sequence of the characteristic sequence of powers of 2
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2的幂特征序列汉克尔行列式的正则性

DOI:
10.1016/j.aam.2018.12.001
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发表时间:
2019-03
影响因子:
1.1
通讯作者:
Ying-Jun Guo
Ying-Jun Guo
中科院分区:
数学3区
文献类型:
--
作者:
Ying-Jun Guo

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对于任意序列u={u(n)} n≥ 0,v={v(n)} n≥ 0,定义了u v:={u(n)v(n)} n≥ 0和u+ v:={u(n)+ v(n)} n≥ 0.我们称整数序列u={u(n)} n≥ 0是多项式生成序列,如果存在整数k≥ 1和多项式fi(x)(0≤ i< k),其系数是整数序列,使得{u(kn + i)} n≥ 0= fi(u),(0≤ i< k).本文研究了多项式生成序列。设k≥ 2且fi(x)= ai x+ B i(0≤ i< k).若ai是k-自动的,B i是k-正则的,且0≤ i< k,则相应的多项式生成序列是k-正则的.作为应用,我们证明了Hankel行列式序列{det <$(p(i+ j))i,j= 0 n− 1} n≥ 0是2-正则的,其中{p(n)} n≥ 0= 0110100010000 <$是2的幂的特征序列.此外,我们还证明了Schueller关于Hankel行列式的一个猜想.
For any sequences u={u (n)} n≥ 0, v={v (n)} n≥ 0, we define u v:={u (n) v (n)} n≥ 0 and u+ v:={u (n)+ v (n)} n≥ 0. We say an integer sequence u={u (n)} n≥ 0 is a polynomial generated sequence if there exists an integer k≥ 1 and polynomials f i (x)(0≤ i< k) whose coefficients are integer sequences such that {u (k n+ i)} n≥ 0= f i (u),(0≤ i< k). In this paper, we study the polynomial generated sequences. Assume k≥ 2 and f i (x)= a i x+ b i (0≤ i< k). If a i are k-automatic and b i are k-regular for 0≤ i< k, then we prove that the corresponding polynomial generated sequences are k-regular. As a application, we prove that the Hankel determinant sequence {det⁡(p (i+ j)) i, j= 0 n− 1} n≥ 0 is 2-regular, where {p (n)} n≥ 0= 0110100010000⋯ is the characteristic sequence of powers of 2. Moreover, we give a proof of a conjecture of Cigler about the Hankel determinants.
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