The Golay-Rudin-Shapiro sequence as “Hankel transform”

Question

Let ${\left( {{a_n}} \right)_{n \geqslant 0}}$ be a sequence of real numbers and ${H_n} = \det \left( {{a_{i + j}}} \right)_{i,j = 0}^{n - 1}$ the $n-$th Hankel determinant. The sequence ${\left( {{H_n}} \right)_{n \geqslant 0}}$ is sometimes called the “Hankel transform” of ${\left( {{a_n}} \right)_{n \geqslant 0}}.$

Is the following result known?

Let $a_0=1$, $a_{2^k-2}=(-1)^k$ for $k\ge2$ and $a_n=0$ else. Then $${\det \left( {{a_{i + j}}} \right)_{i,j = 0}^{n - 1}=r(n)}$$ where ${\left( {{r(n)}} \right)_{n \geqslant 0}}$ is the Golay-Rudin-Shapiro sequence defined by $$\begin{gathered} r(2n) = r(n), \hfill \\ r(2n + 1) = {( - 1)^n}r(n), \hfill \\ r(0) = 1. \hfill \\ \end{gathered} $$

Answer 1

Here is an attempt at analyzing this problem. It seems better to adjust the index of the sequence, so let $\;A_n = b_k \mbox{ if } n=2^k,\; 0\;$ otherwise. Let $\;c_n\; := \det \left( {{A_{i + j}}} \right)_{i,j = 1}^n.\;$ In this determinant as a sum over all permutations, only one permutation in the sum corresponds to a nonzero product of members of the sequence $\{b_k\}.\;$ That is, $c_n = \textrm{sgn}(\sigma(n))\prod_{i=1}^k A_{i+\sigma(n)_i}\;$ where $\;\sigma(n)\;$ is a permutation of $\;\{1,\dots,n\}.\;$ The triangluar array of $\;\{\sigma(n)\}_1^\infty\;$ has a peculiar recursive structure of triangular arrays next to other smaller triangular arrays. In our case, $\;b_1=1,\; b_k=(-1)^k,\;$ for $k>1.\;$ If we know the signature of all $\;\sigma(n),\;$ then the problem is essentially solved.