Explicit, determinantal, recursive formulas and relations of the Peters polynomials and numbers

In this paper, with the help of the Faa di Bruno formula and an identity for the Bell polynomials of the second kind, we find several explicit formulas for the Peters polynomials and numbers. Also, we present determinantal representations for the Peters polynomials and numbers by virtue of a general derivative formula for the ratio of two differentiable functions. Moreover, we give several recursive relations for the Peters polynomials and numbers. As an application, we establish alternative recursive relations for them with the aid of a recursive relation for the Hessenberg determinants. These results cover counterparts for the Boole polynomials and numbers.