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Systems of functional equations. (English) Zbl 0869.39010
In this paper “functional equation” is an equation of particular form from which an implicit unknown function is to be determined. To be exact, the author looks for analytic solutions (complex variables are written in italics, vector variables and vector valued functions with $$n$$ complex components are set in bold face) $${\mathbf y}={\mathbf g}(x,{\mathbf z})$$, with $${\mathbf g}(0,\mathbf{0})=\text\textbf{0}$$, of equations of the form $${\mathbf y}={\mathbf f}(x,{\mathbf y},{\mathbf z})$$. Here $${\mathbf f}$$ is supposed to be analytic in a neighborhood of (0,0,0) with $${\mathbf f}(0,{\mathbf y},{\mathbf z})\equiv \mathbf{0}$$, $${\mathbf f}(x,\mathbf{0},{\mathbf z}) \not\equiv \text\textbf{0}$$ and there exists a component $$y_j$$ of $${\mathbf y}$$ such that the mixed second derivatives of $${\mathbf f}$$ with respect to $$y_j$$ and to any other $$y_k$$ is not identically 0. Furthermore, the Maclaurin coefficients of $${\mathbf f}$$ are nonnegative. Under further assumptions the asymptotic behavior of the Maclaurin coefficients of $${\mathbf g}$$ is determined. Applications to tree enumeration problems and to context-free languages are offered.

##### MSC:
 39B62 Functional inequalities, including subadditivity, convexity, etc. 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX) 05A15 Exact enumeration problems, generating functions 26B10 Implicit function theorems, Jacobians, transformations with several variables 05C30 Enumeration in graph theory 68Q45 Formal languages and automata 39B32 Functional equations for complex functions 41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
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