Spivey's type recurrence relation for degenerate Bell polynomials

TL;DR

Problem: derive Spivey-type recurrences for degenerate Bell polynomials $\phi_{n,\lambda}(x)$ and degenerate $r$-Bell polynomials $\phi_{n,\lambda}^{(r)}(x)$. Approach: apply operator calculus with $X$ and $D$ satisfying $DX-XD=1$ to obtain recurrences in $x$ expressed via $(\cdot)_{n,\lambda}$ and degenerate Stirling numbers. Results: explicit double-sum recurrences for $\phi_{m+n,\lambda}(x)$ and $\phi_{n+m,\lambda}^{(r)}(x)$; with $x=1$ giving the degenerate Bell relations and $\lambda\to0$ recovering the classical Spivey formula. Significance: extends Spivey-type recurrences to degenerate partition-polynomial families via a unified operator-theoretic framework.

Abstract

The aim of this paper is to derive a recurrence relation for the degenerate Bell polynomials by using the operators X and D satisfying the commutation relation DX-XD=1. Here X is the `multiplication by x' operator and D=d/dx. This recurrence relation is a generalization of Spivey's recurrence relation for the Bell numbers. We also obtain a recurrence relation for the degenerate r--Bell polynomials by using the same operators.

Theorems & Definitions (4)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4