A new approach to Bell and poly-Bell numbers and polynomials

Bell polynomials are widely applied in many problems arising from physics and engineering. The aim of this paper is to introduce new types of special polynomials and numbers, namely Bell polynomials and numbers of the second kind and poly-Bell polynomials and numbers of the second kind, and to derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. We also consider degenerate versions of those polynomials and numbers, namely degenerate Bell polynomials and numbers of the second kind and degenerate poly-Bell polynomials and numbers of the second kind, and deduce their similar results.