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

TL;DR

This paper investigates the regularity properties of polynomial generated sequences and proves that certain Hankel determinant sequences, including the characteristic sequence of powers of 2, are regular, providing insights into Cigler's conjecture.

Contribution

It establishes the $k$-regularity of polynomial generated sequences with specific automatic and regular coefficients and confirms Cigler's conjecture for Hankel determinants of the characteristic sequence of powers of 2.

Findings

Hankel determinant sequence of the characteristic sequence of powers of 2 is 2-regular.

Polynomial generated sequences with linear polynomial coefficients are $k$-regular under certain conditions.

Confirmed Cigler's conjecture regarding Hankel determinants.

Abstract

For any sequences $\mathbf{u}=\{u(n)\}_{n\geq0}, \mathbf{v}=\{v(n)\}_{n\geq0},$ we define $\mathbf{u}\mathbf{v}:=\{u(n)v(n)\}_{n\geq0}$ and $\mathbf{u}+\mathbf{v}:=\{u(n)+v(n)\}_{n\geq0}$. Let $f_i(x)~(0\leq i< k)$ be sequence polynomials whose coefficients are integer sequences. We say an integer sequence $\mathbf{u}=\{u(n)\}_{n\geq0}$ is a polynomial generated sequence if $$\{u(kn+i)\}_{n\geq0}=f_i(\mathbf{u}),~(0\leq i< k).$$ %Here we define $\mathbf{u}\mathbf{v}:=\{u(n)v(n)\}_{n\geq0}$ and $\mathbf{u}+\mathbf{v}:=\{u(n)+v(n)\}_{n\geq0}$ for any two sequences $\mathbf{u}=\{u(n)\}_{n\geq0}, \mathbf{v}=\{v(n)\}_{n\geq0}.$ In this paper, we study the polynomial generated sequences. Assume $k\geq2$ and $f_i(x)=\mathbf{a}_ix+\mathbf{b}_i~(0\leq i< k)$. If $\mathbf{a}_i$ are $k$-automatic and $\mathbf{b}_i$ are $k$-regular for $0\leq i< k$, then we prove that the corresponding polynomial generated sequences are $k$-regular. As a application, we prove that the Hankel determinant sequence $\{\det(p_{i+j})_{i,j=0}^{n-1}\}_{n\geq0}$ is $2$-regular, where $\{p(n)\}_{n\geq0}=0110100010000\cdots$ is the characteristic sequence of powers 2. Moreover, we give a answer of Cigler's conjecture about the Hankel determinants.