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 $\mathbf{w}=(w_j{)}_{j\ge 0}$. We investigate Hankel determinants ${\mathcal{H}}_{\mathbf{w}}(n)=\det [s_{\mathbf{w}}(i+j){]}_{0\le i,j<n}$ 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 $t=2\zeta$, where $\zeta$ 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.

arXiv comment37 pages