Various New Properties for Central Bell-Based Type 2 Bernoulli Polynomials of Order β

In recent years, Duran (Fundam. J. Math. Appl., 8(2) (2025), 55-64) introduced the central Bell-based type 2 Bernoulli polynomials of order \( \beta \) given by \( \left( \frac{t}{e^{\frac{t}{2}}-e^{-\frac{t}{2}}}\right) ^{\beta }e^{xt+z\left( e^{\frac{t}{2}}-e^{-\frac{t}{2}}\right) }=\sum_{m=0}^{\infty }% \text{ }_{CB}b_{m}^{\left( \beta \right) }\left( x;z\right) \frac{t^{m}}{m!}% \left( \left\vert t\right\vert <2\pi \right) \) and derived many formulas and relations, covering several symmetric properties, derivative properties, summation formulas, and addition formulas. In this paper, we aim to improve some new properties for the central Bell-based type 2 Bernoulli polynomials of order \( \beta \). We first investigate some new properties, involving central Bell polynomials, classical Bernoulli polynomials and numbers, and central factorial numbers of the second kind. Moreover, we show that the central Bell-based type 2 Bernoulli polynomials of order \( \beta \) are solutions of the some higher-order differential equations. Further, we give a determinantal representation for the central Bell-based type 2 Bernoulli polynomials of order \( \beta \).