Recurrence relations for degenerate Bell and Dowling polynomials via Boson operators

TL;DR

The paper addresses the problem of extending Spivey’s recurrence, originally for Bell numbers, to degenerate Bell and Dowling polynomials using a bosonic operator framework. It employs normal ordering of degenerate powers of number operators with boson operators satisfying $[a,a^{\dagger}]=1$ to derive explicit recurrences, expressing results in terms of degenerate Stirling and Whitney numbers. The key contributions are the Spivey-type recurrences $\phi_{n+m,\lambda}(x)=\sum_{j=0}^{m}\sum_{k=0}^{n}{m \brace j}_{\lambda}\binom{n}{k}(j-m\lambda)_{n-k,\lambda}x^{j}\phi_{k,\lambda}(x)$ and the analogous formulas for $D_{m,\lambda}(n,x)$ and $D_{m,\lambda}^{(r)}(n,x)$, plus reduced forms as $\lambda\to 0$ that recover the classical identities. This work links degenerate combinatorial polynomials to operator methods, providing new tools for analysis and potential applications in quantum-operator contexts. The results unify and extend known recurrences within a coherent degenerate framework.

Abstract

Spivey found a recurrence relation for the Bell numbers by using combinatorial method. The aim of this paper is to derive Spivey's type recurrence relations for the degenerate Bell polynomials and the degenerate Dowling polynomials by using the boson annihilation and creation operators satisfying the commutation relation aa+-a+a=1. In addition, we derive a Spivey's type recurrence relation for the r-Dowling polynomials.

Theorems & Definitions (7)

• Proposition 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 3.1
• Theorem 3.2