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$B$-Stirling numbers associated to potential polynomials

José A. Adell, Beáta Bényi

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.

$B$-Stirling numbers associated to potential polynomials

Abstract

We introduce the -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 -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.

Paper Structure

This paper contains 8 sections, 9 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. $B$-Stirling numbers
  3. Inner operations in the set ${\mathcal{B}}$
  4. Examples
  5. Example 1. Partial and complete Bell polynomials
  6. Example 2. Degenerate Stirling numbers
  7. Example 3. Probabilistic Stirling numbers
  8. Example 4. $S$-restricted Stirling numbers

Key Result

Theorem 1

Let $B(z)\in {\mathcal{B}}$. Then,

Theorems & Definitions (22)

  • Theorem 1
  • proof
  • Remark 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Remark 5
  • Theorem 6
  • proof
  • ...and 12 more