Central Bell-Based Type 2 Bernoulli Polynomials of Order $\beta $

In recent years, Hermite-based special polynomials, Bell-based special polynomials, and Laguerre-based special polynomials have been explored, and numerous properties and applications have been investigated by many mathematicians. Here, we consider the central Bell-based type 2 Bernoulli polynomials of order $\beta $ that extend the concepts of central Bell polynomials and type 2 Bernoulli polynomials. Then, we derive diverse formulas, relations, and identities, such as some summation formulas, an addition formula, two partial derivative properties, a recurrence relation, two explicit formulas, and two summation formulas covering central Bell polynomials and central factorial numbers of the second kind. Moreover, we investigate an implicit summation formula for central Bell-based type 2 Bernoulli polynomials of order $\beta $ utilizing some series manipulation methods. Also, we developed three useful symmetric identities for the central Bell-based type 2 Bernoulli polynomials of order $\beta $.