Stirling Polynomials in Several Indeterminates

The classical exponential polynomials, today commonly named after
E. ,T. Bell, have a wide range of remarkable applications in
Combinatorics, Algebra, Analysis, and Mathematical Physics. Within the
algebraic framework presented in this book they appear as structural
coefficients in finite expansions of certain higher-order derivative
operators. In this way, a correspondence between polynomials and
functions is established, which leads (via compositional inversion) to
the specification and the effective computation of orthogonal
companions of the Bell polynomials. Together with the latter, one
obtains the larger class of multivariate `Stirling polynomials'. Their
fundamental recurrences and inverse relations are examined in detail
and shown to be directly related to corresponding identities for the
Stirling numbers. The following topics are also covered: polynomial
families that can be represented by Bell polynomials; inversion
formulas, in particular of Schlömilch-Schläfli type; applications to
binomial sequences; new aspects of the Lagrange inversion, and, as a
highlight, reciprocity laws, which unite a polynomial family and that
of orthogonal companions. Besides a
textsl{Mathematica, textregistered package and an extensive
bibliography, additional material is compiled in a number of notes and
supplements.