A Generalized Recurrence for fully degenerate Bell polynomials

TL;DR

This work resolves an issue in Spivey-type recurrences for fully degenerate Bell polynomials by introducing new polynomials $B_{n,\lambda}(x)$ and their two-variable counterparts $B_{n,\lambda}(x,y)$, as well as their fully degenerate $r$-counterparts $B_{n,\lambda}^{(r)}(x)$ and $B_{n,\lambda}^{(r)}(x,y)$. Employing the operator pair $(X,D)$ with $DX-XD=1$, the authors derive natural Spivey-type recurrence relations, along with Dobinski-like formulas, finite-sum expressions, and operator representations for these polynomials and their variants. The results extend to two-variable cases and $r$-Bell polynomials, providing a cohesive framework that reduces to classical Bell polynomials in the limit $\lambda\to 0$. This framework lays groundwork for further algebraic and combinatorial exploration of degenerate polynomial families and their interrelations.

Abstract

This paper addresses the unnatural appearance of the two-variable degenerate Fubini polynomials in a recently derived Spivey-type recurrence relation for the fully degenerate Bell polynomials. To solve this, we introduce a new family of polynomial which we also call the fully degenerate Bell polynomials, along with their two-variable counterparts. Our main contribution is the derivation of natural Spivey-type recurrence relations using operator methods. We extend these results to the r-counterparts, the fully degenerate r-Bell polynomials providing Dobinski-like, finite sum, operator expressions, and Spivey-type recurrence relations for all the new polynomials.

Theorems & Definitions (16)

• Theorem 2.1
• Remark 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8
• Theorem 2.9
• Theorem 2.10