A Nonlinear Differential Operator Series That Commutes with Any Function

A natural differential operator series is one that commutes with every function. The only linear examples are the formal series operators $e^{\alpha zD} $ representing translations. This paper discusses a surprising natural nonlinear “normally ordered” differential operator series, arising from the Lagrange inversion formula. The operator provides a wide range of new higher-order derivative identities and identities among Bell polynomials. These identities specialize to a large variety of interesting identities among binomial coefficients and classical orthogonal polynomials, a number of which are new.