Tauberian Theorems for Sequences linked by a Convolution

Given sequences g, λ: IN₀ → C, g (0) = 1, linked by the convolution λ g = g′¹ (g′¹(n)= (n + 1) g (n + 1)) we study what can be inferred about λ (n → ∞) from some concrete information about the behaviour of g (n → ∞).