156 research outputs found
Infinite products involving binary digit sums
Let denote the Thue-Morse sequence with values . The
Woods-Robbins identity below and several of its generalisations are well-known
in the literature
\begin{equation*}\label{WR}\prod_{n=0}^\infty\left(\frac{2n+1}{2n+2}\right)^{u_n}=\frac{1}{\sqrt
2}.\end{equation*} No other such product involving a rational function in
and the sequence seems to be known in closed form. To understand these
products in detail we study the function
\begin{equation*}f(b,c)=\prod_{n=1}^\infty\left(\frac{n+b}{n+c}\right)^{u_n}.\end{equation*}
We prove some analytical properties of . We also obtain some new identities
similar to the Woods-Robbins product.Comment: Accepted in Proc. AMMCS 2017, updated according to the referees'
comment
Extremal properties of (epi)Sturmian sequences and distribution modulo 1
Starting from a study of Y. Bugeaud and A. Dubickas (2005) on a question in
distribution of real numbers modulo 1 via combinatorics on words, we survey
some combinatorial properties of (epi)Sturmian sequences and distribution
modulo 1 in connection to their work. In particular we focus on extremal
properties of (epi)Sturmian sequences, some of which have been rediscovered
several times
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