Moments of a Thue-Morse generating function
This article contains results on the moments \[ M_k(n)=\int_0^1\, |T_n(x)|^{2k}\, dx\quad (k\geq 1) \] of the generating function \[ T_n(x)=\prod_{0\leq r< n}\, (1-e(2^r x)), \] employing the notation \(e(\theta)=e^{2\pi i\theta}\). It is revealed that \(M_k(n)\) satisfies a linear recurrence of order \(k\) given by \[ M_k(n)=c_{k-1,k}M_k(n-1)+\cdots + c_{0,k}M_k(n-k), \] where these integral coefficients come from the characteristic polynomial \[ p_k(z)=z^k-c_{k-1,k}z^{k-1}-\cdots - c_{0,k} \] belonging to the \(k\times k\) matrix \(A_k=[a_{ij}]\) defined by \[ a_{ij}=(-1)^i \;2^{2k-2i-1}\binom {j+k-i}{i} \frac{j+k}{j+k-i} \quad (0\leq i,j< k). \] A recursive property of the traces of \(A_k\) is also given, following some results involving the spectral radius \(\rho(A_k)\) and its upper bound [Theorems 2, 3, 4]. A stronger upper bound for \(\rho(A_k)\) is predicted, along with several other interesting conjectures concerning the characteristic polynomial and eigenvalues of \(A_k\) [Section 7]. It is noted that the function \(T_n(x)\) being considered has had many fruitful roles in different problems and is associated with the Prouhet-Thue-Morse sequence via the relation \[ T_n(x)=\sum_{0\leq m<2^n}\, (-1)^{s(m)} e(mx), \] where \(s(m)\) is the base-2 digit sum of \(m\). Meanwhile, the Prouhet-Thue-Morse sequence -- here appearing in the form \((-1)^{s(m)}\) -- is itself a widespread phenomenon found across diverse fields of study, surprisingly connecting otherwise unrelated topics such as \(\beta\)-expansions and an infinite game of chess. For a stimulating introduction, the first item listed in the bibliography [\textit{J.-P. Allouche} and \textit{J. Shallit}, in: Sequences and their applications. Proceedings of the international conference, SETA '98, Singapore, December 14--17, 1998. London: Springer. 1--16 (1999; Zbl 1005.11005)] is suitably a first to read.
scientific article; zbMATH DE number 6919690