On the generalization of the Lambert $W$ function with applications in theoretical physics

The Lambert $W$ function, giving the solutions of a simple transcendental equation, has become a famous function and arises in many applications in combinatorics, physics, or population dyamics just to mention a few. In the last decade it turned out that some specific relativistic equations and molecular physics problems need solutions of more general transcendental equations. Thus a generalization of the Lambert function is necessary. In this paper we construct this generalization which involves some special polynomials.