Hankel determinants of weighted binary sums of digits

Abstract

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.