Functional equations with multiple recursive terms

Abstract In this paper, we study a functional equation for generating functions of the form $$f(z) = g(z) \sum _{i=1}^M p_i f(\alpha _i(z)) + K(z)$$ f ( z ) = g ∑ i 1 M p α + K , viz. recursion with multiple recursive terms. We derive and analyze solution case that $$\alpha _i(z)$$ are commutative contraction mappings. The results applied to wide range queueing, autoregressive branching processes.