Nonlinear Inverse Relations for Bell Polynomials via the Lagrange Inversion Formula

In this paper, by using the classical Lagrange inversion formula, we establish a nonlinear inverse relation that involves the Bell polynomials. As applications of this inverse relation, we not only find a short proof of another nonlinear inverse relation due to Birmajer et al., but also deduce some allied combinatorial identities. Finally, we propose the general problem of finding nonlinear inverse relations and give a positive solution to it.