Schreiber A. Stirling Polynomials in Several Indeterminates

Berlin: Logos Verlag, 2021. — 164 p.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 Schlomilch-Schlafli 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 Mathematica(R) package and an extensive bibliography, additional material is compiled in a number of notes and supplements.Multivariate Stirling PolynomialsFunction algebra with derivation
Expansion of higher-order derivatives
A brief summary on Bell polynomials
Inversion formulas and recurrences
Explicit formulas for Sn;k
Remarks on Lagrange inversion
Concluding remarks
Inverse Relations and Reciprocity LawsBasic notions and preliminaries
Polynomials from Taylor coeXcients
Composition rules
Representation by Bell polynomials
Applications to binomial sequences
Lagrange inversion polynomials
Reciprocity theorems
Appendix A Mathematica Package