Bell-Based Bernoulli Polynomials with Applications

In this paper, we consider Bell-based Stirling polynomials of the second kind and derive some useful relations and properties including some summation formulas related to the Bell polynomials and Stirling numbers of the second kind. Then, we introduce Bell-based Bernoulli polynomials of order $\alpha$ and investigate multifarious correlations and formulas including some summation formulas and derivative properties. Also, we acquire diverse implicit summation formulas and symmetric identities for Bell-based Bernoulli polynomials of order $\alpha$. Moreover, we attain several interesting formulas of Bell-based Bernoulli polynomials of order $\alpha$ arising from umbral calculus.