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

We evaluate in closed form series of the type ∑ 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: ∑ n>1 (−1)2 4n+ 1 2n(2n+ 1)(2n+ 2) = − 4 where s2(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 > 0: ∑ n>0 v(n) (n+ 1)2d+1 = π|E2d| (22d+2 − 2)(2d)! where (v(n))n is the ±1 regular paperfolding sequence and E2d is an Euler number.