Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 90, pp. 1-32.

Title: New approach to the Lagrange-B\"urmann theorem via omega calculus and applications

Author: Antonio Francisco Neto (UFOP, Morro do Cruzeiro, Brazil)

Abstract:
A novel approach to the ubiquitous multidimensional Lagrange-Burmann Theorem is developed 
which uses the omega calculus (OC) developed long ago by MacMahon to study the partition of
natural numbers. Several applications are given including the answer to open questions
regarding the generalized Lambert function $W$ as stated in [56].
More precisely, a master theorem is presented introducing a new generalized Lambert
function for which several previously known representations arise as special cases
including most Taylor series results of [56] and some other integral representations.
Furthermore, the convergence radius of the aforementioned generalized Lambert
function is explicitly determined for which even special cases were not known before.
This work shows another instance where omega calculus is useful, this time,
to address inverse problems of general interest related to functional equations.

Submitted June 25, 2025. Published September 26, 2025.
Math Subject Classifications: 05A15, 05A17, 34K05, 44A99.
Key Words: Lagrange-Burmann theorem; Omega calculus; Lambert function; functional equation; inverse problem.