Spivey's type recurrence relation for Lah-Bell polynomials

TL;DR

The paper addresses extending Spivey's recurrence to Lah-Bell polynomials and their $r$- and $\lambda$-analogues. It employs an operator approach with $X$ and $D$ satisfying $DX-XD=1$ to derive a double-sum recurrence for $LB_{n+m}^{(r)}(x)$ and related explicit coefficients, and provides an alternative method for the $\lambda$-analogue. Key contributions include an explicit formula $L^{r}(n,k)=\frac{n!}{k!}\binom{n+r-1}{k+r-1}$, Spivey-type recurrences for $LB_{n+m}^{(r)}(x)$ and $LB_{n+m,\lambda}^{(r)}(x)$, and generating functions with Dobinski-like representations for the $r$-Lah and its degenerate ($\lambda$) analogue. The results extend combinatorial partition polynomials and furnish analytic tools (generating functions, closed forms) for studying Lah-Bell families and their parametrized variants.

Abstract

The aim of this paper is to derive Spivey's type recurrence relations for the Lah-Bell polynomials and the r-Lah-Bell polynomials by utilizing operators X and D satisfying the commutation relation DX-XD=1. Here X is the `multiplication by x' operator and D is the differentiation operator D=d/dx. In addition, we obtain Spivey's type recurrence relation for the lambda analogue of r-Lah-Bell polynomials by some other method without using the operators X and D.

Theorems & Definitions (10)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4