On a Novel Class of Special Polynomials: Central Bell-Based Type 2 Euler Polynomials Associated with Umbral Calculus

Abstract

In this work, the authors define the central Bell-based central factorial polynomials of the second kind and examine many of their fundamental formulae, properties, and relations, some of which are derived using the umbral calculus technique. Then, the authors introduce the central Bell-based type 2 Euler polynomials of order α that extend the concepts of central Bell polynomials and type 2 Euler polynomials. For these polynomials, the authors 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 factorial numbers of the second kind and central Bell polynomials. Moreover, the authors investigate two implicit summation formulas for central Bell-based type 2 Euler polynomials of order α utilizing some series manipulation methods. Also, the authors develop useful identities of symmetry for the central Bell-based type 2 Euler polynomials of order α. Furthermore, the authors obtain several interesting formulas of the central Bell-based type 2 Euler polynomials of order α arising from umbral calculus to possess alternative ways of obtaining our consequences and also some new consequences. Finally, the authors provide a determinantal representation for central Bell-based type 2 Euler polynomials.