Summation of Rational Series Twisted by Strongly $B$-multiplicative Coefficients

Jean-Paul Allouche, Jonathan Sondow

Abstract


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.


Keywords


Summation of series; Strongly $B$-multiplicative sequences; Paperfolding sequence; Shapiro-Rudin sequence.

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