Study on Nonlinear Inverse Relations of Bell Polynomials via the Lagrange Inversion Formula.

In this paper, by using the classical Lagrange inversion formula, we solve a general problem of finding nonlinear inverse relations proposed in the previous paper [J. Wang, Nonlinear inverse relations for Bell polynomials via the Lagrange inversion formula, J. Int. Seq., Vol. 22 (2019), Article 19.3.8. https://cs.uwaterloo.ca/journals/JIS/VOL22/Wang/wang53.pdf]. 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 set up a few convolution identities concerning the Mina polynomials.