On The Improvement of Combinatorial Mathematics Teaching From Generating Function

Generating function is an effective method to solve combinatorial counting problem, but it most likely to be neglected in combinatorial mathematics teaching. In this paper, we provides a demonstration for combinatorial mathematics teaching improvement by using the generating function solving combination number and the sum of preceding terms among sequence of numbers.

Square root singularities of infinite systems of functional equations

International audience 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 $\mathbf{y} (x)$.