Multivariate Stirling Polynomials of the first and second kind

Abstract: Two doubly indexed families of homogeneous and isobaric polynomials in several indeterminates are considered: the (partial) exponential Bell polynomials $B_{n,k}$ and a new family $S_{n,k}$ such that $X_1^{-(2n-1)}S_{n,k}$ and $B_{n,k}$ obey an inversion law which generalizes that of the Stirling numbers of the first and second kind. Both polynomial families appear as Lie coefficients in expansions of certain derivatives of higher order. Substituting $D^j(\varphi)$ (the $j$-th derivative of a fixed function $\varphi$) in place of the indeterminates $X_j$ shows that both $S_{n,k}$ and $B_{n,k}$ are differential polynomials depending on $\varphi$ and on its inverse $\overline{\varphi}$, respectively. Some new light is shed thereby on Comtet's solution of the Lagrange inversion problem in terms of the Bell polynomials. According to Haiman and Schmitt that solution is essentially the antipode on the Fa`{a} di Bruno Hopf algebra. It can be represented by $X_1^{-(2n-1)}S_{n,1}$. Moreover, a general expansion formula that holds for the whole family $S_{n,k}$ ($1\leq k\leq n$) is established together with a closed expression for the coefficients of $S_{n,k}$. Several important properties of the Stirling numbers are demonstrated to be special cases of relations between the corresponding polynomials. As a non-trivial example, a Schl\"omilch-type formula is derived expressing $S_{n,k}$ in terms of the Bell polynomials $B_{n,k}$, and vice versa.