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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 this paper we construct and study in great detail a generalization of the Lambert W which involves some special polynomials and even combinatorial aspects.doi:10.1090/tran/6911 fatcat:z6fxguaznfe45lj4tcwtoksgjq