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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>r</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msubsup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> ${B}_{n,\lambda }^{(r)}(x)$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>r</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msubsup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> ${B}_{n,\lambda }^{(r)}(x,y)$ , providing Dobinski-like, finite sum, operator expressions, and Spivey-type recurrence relations for all the new polynomials.