Transcendence of infinite series over lattices

In this paper, we study the arithmetic nature of series of the form ∑ ω ∈ Λ A ( ω ) B ( ω ) , where Λ is a two-dimensional lattice in C , A ( X ) and B ( X ) are suitable polynomials over C , with deg A ≤ deg B - 3 . In particular, we focus on the cases when the roots of the polynomial B ( X ) are either algebraic numbers or rational multiples of a non-zero period of Λ .