We introduce a linear infinitary λ-calculus, called Λ∞, in which two exponential modalities are available, the first one being the usual, finitary one, the other being the only construct interpreted coinductively. The obtained calculus embeds the infinitary applica-tive λ-calculus and is universal for computations over infinite strings. What is particularly interesting about Λ∞, is that the refinement induced by linear logic allows to restrict both modalities so as to get calculi which are terminating inductively and productive coin-ductively. We exemplify this idea by analysing a fragment of Λ built around the principles of SLL and 4LL. Interestingly, it enjoys confluence, contrarily to what happens in ordinary infinitary λ-calculi.