4,786 research outputs found

### Infinite products involving binary digit sums

Let $(u_n)_{n\ge 0}$ denote the Thue-Morse sequence with values $\pm 1$. 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 $n$
and the sequence $u_n$ 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 $f$. We also obtain some new identities
similar to the Woods-Robbins product.Comment: Accepted in Proc. AMMCS 2017, updated according to the referees'
comment

### Summation of Series Defined by Counting Blocks of Digits

We discuss the summation of certain series defined by counting blocks of
digits in the $B$-ary expansion of an integer. For example, if $s_2(n)$ denotes
the sum of the base-2 digits of $n$, we show that $\sum_{n \geq 1}
s_2(n)/(2n(2n+1)) = (\gamma + \log \frac{4}{\pi})/2$. We recover this previous
result of Sondow in math.NT/0508042 and provide several generalizations.Comment: 12 pages, Introduction expanded, references added, accepted by J.
Number Theor

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