We evaluate in closed form series of the type $\sum u(n) R(n)$, with $(u(n))_n$ a strongly $B$-multiplicative sequence and $R(n)$ a (well-chosen) rational function. A typical example is:$$\sum_{n \geq 1} (-1)^{s_2(n)} \frac{4n+1}{2n(2n+1)(2n+2)} = -\frac{1}{4}$$where $s_2(n)$ is the sum of the binary digits of the integer $n$. Furthermore closed formulas for series involving automatic sequences that are not strongly $B$-multiplicative, such as the regular paperfolding and Golay-Shapiro-Rudin sequences, are obtained; for example, for integer $d \geq 0$:$$\sum_{n \geq 0} \frac{v(n)}{(n+1)^{2d+1}} = \frac{\pi^{2d+1} |E_{2d}|}{(2^{2d+2}-2)(2d)!}$$where $(v(n))_n$ is the $\pm 1$ regular paperfolding sequence and $E_{2d}$ is an Euler number.
@article{10_37236_4630,
author = {Jean-Paul Allouche and Jonathan Sondow},
title = {Summation of rational series twisted by strongly {\(B\)-multiplicative} coefficients},
journal = {The electronic journal of combinatorics},
year = {2015},
volume = {22},
number = {1},
doi = {10.37236/4630},
zbl = {1307.11014},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4630/}
}
TY - JOUR
AU - Jean-Paul Allouche
AU - Jonathan Sondow
TI - Summation of rational series twisted by strongly \(B\)-multiplicative coefficients
JO - The electronic journal of combinatorics
PY - 2015
VL - 22
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/4630/
DO - 10.37236/4630
ID - 10_37236_4630
ER -
%0 Journal Article
%A Jean-Paul Allouche
%A Jonathan Sondow
%T Summation of rational series twisted by strongly \(B\)-multiplicative coefficients
%J The electronic journal of combinatorics
%D 2015
%V 22
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/4630/
%R 10.37236/4630
%F 10_37236_4630
Jean-Paul Allouche; Jonathan Sondow. Summation of rational series twisted by strongly \(B\)-multiplicative coefficients. The electronic journal of combinatorics, Tome 22 (2015) no. 1. doi: 10.37236/4630
