New approach to the Lagrange-Burmann theorem via omega calculus and applications

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. For more information and the Latex file see https://ejde.math.txstate.edu/Volumes/2025/90/abstr.html