Factorial Polynomials and Associated Number Families

TL;DR

This paper explores two families of factorial polynomials in multiple variables, examining their relationships to falling and rising factorials, and investigates their inversion relations and associated number families.

Contribution

It introduces and analyzes two new doubly indexed polynomial families related to factorials, extending the theory of potential polynomials and their inversion properties.

Findings

Established inversion relations for the factorial polynomial families

Identified new associated number families linked to these polynomials

Extended the framework of potential polynomials to multivariate factorial polynomials

Abstract

Two doubly indexed families of polynomials in several indeterminates are considered. They are related to the falling and rising factorials in a similar way as the potential polynomials (introduced by L. Comtet) are related to the ordinary power function. We study the inversion relations valid for these factorial polynomials as well as the number families associated with them.