The Fibonacci sequence and an elliptic curve

Infinite series involving the reciprocal Fibonacci numbers may ad-mit no algebraic dependence between each other over the rational numbers. In this note, we introduce an identity which reveals an algebraic dependence relation between two infinite series involving the reciprocal Fibonacci num-bers. The identity was discovered from a peculiar description of an elliptic function, and this observation is generalized to produce similar identities on a large class of sequences defined by linear recurrences on three consecutive terms.