Insight into degenerate Bell-based Bernoulli polynomials with applications

Recently, the Bell-based Stirling polynomials of the second kind and the Bell-based Bernoulli polynomials [{U. Duran, S. Araci, M. Acikgoz, Axioms, \textbf{10} (2021), 23 pages}] have been considered, and some of their properties and applications in umbral calculus have been derived and analyzed. In this work, a degenerate form of the Bell-based Stirling polynomials of the second kind is defined, and several fundamental properties and formulas for these polynomials are investigated and presented in detail. Then, a degenerate form of the Bell-based Bernoulli polynomials of order \(\alpha \) is defined and a plenty of their properties\ are examined in different aspects. Several correlations with other polynomials and numbers in literature, symmetric identities, implicit summation formulas, derivative properties and addition formulas for the mentioned new polynomials are derived in detail, and some special cases of these results are investigated. Also, the degenerate Bell-based Bernoulli polynomials of order \(\varepsilon \) are studied in \(\lambda \)-umbral calculus and interesting relations and formulas are developed. Furthermore, the application of \(\lambda \)-umbral calculus to Bell-based degenerate Bernoulli polynomials of order \(\varepsilon \) shows a correlation with higher-order degenerate derangement polynomials. Finally, a representation of the degenerate differential operator on the degenerate Bell-based Bernoulli polynomials of order \(\varepsilon \) is provided.