$B$-Stirling numbers associated to potential polynomials

TL;DR

This paper introduces the $B$-Stirling numbers, a new family of combinatorial coefficients linked to potential polynomials, unifying various known Stirling and Bell numbers, with recursive computation methods.

Contribution

The paper defines $B$-Stirling numbers, explores their properties, and provides recursive formulas and relations to classical Stirling numbers for efficient computation.

Findings

$B$-Stirling numbers generalize many known Stirling and Bell numbers.

Recursive formulas enable efficient computation of these numbers.

Relations between $B$-Stirling numbers and classical Stirling numbers are established.

Abstract

We introduce the $B$-Stirling numbers of the first and second kind, which are the coefficients of the potential polynomials when we express them in terms of the monomials and the falling factorials, respectively. These numbers include, as particular cases, the partial and complete Bell polynomials, the degenerate and probabilistic Stirling numbers, and the $S$-restricted Stirling numbers, among others. Special attention is devoted to the computation of such numbers. On the one hand, a recursive formula is provided. On the other, we can compute Stirling numbers of one kind in terms of the other, with the help of the classical Stirling numbers.