A generalized recurrence for fully degenerate Bell polynomials

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 Bel n , λ ( x ). To solve this, we introduce a new family of polynomials, denoted B n , λ ( x ), which we also call the fully degenerate Bell polynomials, along with their two-variable counterparts, B n , λ ( x , y ). Our main contribution is the derivation of natural Spivey-type recurrence relations for B n , λ ( x ) and B n , λ ( x , y ) using operator methods. We extend these results to the r -counterparts, the fully degenerate r -Bell polynomials B n , λ ( r ) ( x ) ${B}_{n,\lambda }^{(r)}(x)$ and B n , λ ( r ) ( x , y ) ${B}_{n,\lambda }^{(r)}(x,y)$ , providing Dobinski-like, finite sum, operator expressions, and Spivey-type recurrence relations for all the new polynomials.