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 Bn,k and a new family Sn,k∈Z[X1,…,Xn−k+1] such that X1−(2n−1)Sn,k and Bn,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 Dj(φ) (the jth derivative of a fixed function φ) in place of the indeterminates Xj shows that both Sn,k and Bn,k are differential polynomials depending on φ and on its inverse φ¯, 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 Faà di Bruno Hopf algebra. It can be represented by X1−(2n−1)Sn,1. Moreover, a general expansion formula that holds for the whole family Sn,k (1≤k≤n) is established together with a closed expression for the coefficients of Sn,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 Schlömilch-type formula is derived expressing Sn,k in terms of the Bell polynomials Bn,k, and vice versa.