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

TL;DR

This paper explores how fixed point theorems and iterative methods can be applied to combinatorial mathematics, providing new proofs and algorithms for the Lagrange Inversion Formula related to Riordan arrays.

Contribution

It introduces a novel approach using Banach Fixed Point Theorem to derive proofs and algorithms for the Lagrange Inversion Formula in combinatorics.

Findings

Provided a new proof of the Lagrange Inversion Formula

Developed an iterative algorithm based on fixed point methods

Linked fixed point theory with Riordan array transformations

Abstract

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.