Twisted rational functions and series

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...

Rational series for multiple zeta and log gamma functions

We give series expansions for the Barnes multiple zeta functions in terms of rational functions whose numerators are complex-order Bernoulli polynomials, and whose denominators are linear. We also derive corresponding rational expansions for Dirichlet L-functions and multiple log gamma functions in terms of higher order Bernoulli polynomials. These expansions naturally express many of the well-...

Non-commutative Rational Power Series and Algebraic Generating Functions

Sequences of numbers abound in combinatorics whose generating functions are algebraic over the rational functions. Examples include Catalan and related numbers, numbers of words expressing an element in a free group, and diagonal coe cients of 2-variable rational generating functions (Furstenberg's theorem). Algebraicity is of of practical as well as theoretical interest, since it guarantees an...

Twisted L-Functions and Monodromy

On Rational Series and Rational Languages

We study the connections between rational series with coeecients in a semiring and their languages.