Johannes F. Morgenbesser

Square root singularities of infinite systems of functional equations
dmtcs:2782 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2010,
DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10)

https://doi.org/10.46298/dmtcs.2782
Square root singularities of infinite systems of functional equationsArticle
Authors: Johannes F. Morgenbesser ^{1}
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Johannes F. Morgenbesser
1 Institut für Diskrete Mathematik und Geometrie [Wien]
Infinite systems of equations appear naturally in combinatorial counting problems. Formally, we consider functional equations of the form $\mathbf{y} (x)=F(x,\mathbf{y} (x))$, where $F(x,\mathbf{y} ):\mathbb{C} \times \ell^p \to \ell^p$ is a positive and nonlinear function, and analyze the behavior of the solution $\mathbf{y} (x)$ at the boundary of the domain of convergence. In contrast to the finite dimensional case different types of singularities are possible. We show that if the Jacobian operator of the function $F$ is compact, then the occurring singularities are of square root type, as it is in the finite dimensional setting. This leads to asymptotic expansions of the Taylor coefficients of $\mathbf{y} (x)$.
Volume: DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10)