Recurrence relations for degenerate Bell and Dowling polynomials via boson operators

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.