AI 中文总结
研究加权二进制数字和的汉克尔行列式\(\mathcal{H}_\mathbf{w}(n)\)，推导递归式计算它，给出普通二进制数字和时的闭式解，还研究\(w_j = t^j\)时情况及相关一阶差分的行列式，推广了倍周期序列汉克尔行列式的结果。
AI 中文摘要
设\(s_\mathbf{w}\)是与任意复权重序列\(\mathbf{w}=(w_j)_{j\geq 0}\)相关的加权二进制数字和函数。研究汉克尔行列式\(\mathcal{H}_\mathbf{w}(n) = \det [s_{\mathbf{w}}(i + j)]_{0\leq i,j < n}\)，推导通用递归式以有效计算\(\mathcal{H}_\mathbf{w}(n)\)。应用于普通二进制数字和时，给出多个指标序列下\(\mathcal{H}_\mathbf{w}(n)\)的闭式，解决了Allouche和Shallit提出的部分问题。还研究了\(w_j = t^j\)时行列式的情况，探讨其零点。对于\(t = 2\zeta\)（\(\zeta\)为单位根），行列式在大的结构化指标集上为零，其余部分稀疏但无限。此外，考虑与\(s_{\mathbf{w}}\)一阶差分相关的汉克尔行列式，得到显式乘积公式，推广了Fokkink、Kraaikamp和Shallit关于倍周期序列汉克尔行列式的结果。
英文摘要
Let $s_\mathbf{w}$ be the weighted binary sum-of-digits function associated with an arbitrary sequence of complex weights $\mathbf{w}=(w_j)_{j\geq 0}$. We investigate Hankel determinants $\mathcal{H}_\mathbf{w}(n) = \det [s_{\mathbf{w}}(i+j)]_{0\leq i,j<n}$ and derive a general recursion that allows us to effectively compute $\mathcal{H}_\mathbf{w}(n)$ for all $n$. Applying it to the ordinary binary sum-of-digits, that is, $w_j=1$, we express $\mathcal{H}_\mathbf{w}(n)$ in a closed form for several sequences of indices, including the remarkably simple $$ \mathcal{H}_\mathbf{w}(\lceil 2^{k+2}/3\rceil)= (-1)^{\frac{(k+2)(k+3)}{2}}(k+1). $$ This yields an infinite family of explicit evaluations, giving a partial solution to a problem posed by Allouche and Shallit. Moreover, we closely study the specialization $w_j=t^j$, where the determinants become polynomials in $t$, and investigate their vanishing. For $t=2ζ$, where $ζ$ is a root of unity, we show that the determinants vanish on a large structured set of indices, while the complementary is sparse but infinite. In addition to $\mathcal{H}_\mathbf{w}(n)$, we consider Hankel determinants associated with the first difference of $s_{\mathbf{w}}$, obtaining an explicit product formula. This generalizes the results by Fokkink, Kraaikamp, and Shallit concerning Hankel determinants for the period-doubling sequence.
Comments37 pages
