Iterative Processes Related to Riordan Arrays: The Reciprocation and the Inversion of Power Series

We point out how Banach Fixed Point Theorem, and the Picard successive approximation methods induced by it, allows us to treat some mathematical methods in Combinatorics. In particular we get, by this way, a proof and an iterative algorithm for the Lagrange Inversion Formula.Comment: 17 pages. We extend the results in the previuous version proving finally the Lagrange Inversion Formula via Banach Fixed Point Theore

A Residue Theorem for Malcev-Neumann Series

In this paper, we establish a residue theorem for Malcev-Neumann series that requires few constraints, and includes previously known combinatorial residue theorems as special cases. Our residue theorem identifies the residues of two formal series that are related by a change of variables. We obtain simple conditions for when a change of variables is possible, and find that the two related formal series in fact belong to two different fields of Malcev-Neumann series. The multivariate Lagrange inversion formula is easily derived and Dyson's conjecture is given a new proof and generalized.Comment: 22 pages, extensive revisio

The Lagrange inversion formula on non-Archimedean fields. Non-Analytical Form of Differential and Finite Difference Equations

The classical Lagrange inversion formula is extended to analytic and non--analytic inversion problems on non--Archimedean fields. We give some applications to the field of formal Laurent series in $n$ variables, where the non--analytic inversion formula gives explicit formal solutions of general semilinear differential and $q$--difference equations. We will be interested in linearization problems for germs of diffeomorphisms (Siegel center problem) and vector fields. In addition to analytic results, we give sufficient condition for the linearization to belong to some Classes of ultradifferentiable germs, closed under composition and derivation, including Gevrey Classes. We prove that Bruno's condition is sufficient for the linearization to belong to the same Class of the germ, whereas new conditions weaker than Bruno's one are introduced if one allows the linearization to be less regular than the germ. This generalizes to dimension $n> 1$ some results of [CarlettiMarmi]. Our formulation of the Lagrange inversion formula by mean of trees, allows us to point out the strong similarities existing between the two linearization problems, formulated (essentially) with the same functional equation. For analytic vector fields of \C^2 we prove a quantitative estimate of a previous qualitative result of [MatteiMoussu] and we compare it with a result of [YoccozPerezMarco].Comment: This is the final version in press on DCDS Series A. Some minor changes have been made, in particular the relation w.r.t. the results of Perez Marco and Yocco

Non-commutative extensions of the MacMahon Master Theorem

We present several non-commutative extensions of the MacMahon Master Theorem, further extending the results of Cartier-Foata and Garoufalidis-Le-Zeilberger. The proofs are combinatorial and new even in the classical cases. We also give applications to the $\beta$-extension and Krattenthaler-Schlosser's $q$-analogue.Comment: 28 pages, 6 figure