New Formulas for Apostol‐Type Bernoulli Polynomials of Complex Order: Approach to Derivative Operator and Functional Equation Method

ABSTRACT The aim of this paper is to derive new formulas for Apostol‐type Bernoulli polynomials and numbers of complex order by using derivative operators and the functional equation method. Applying the derivative operator and the Euler derivative operator to the corresponding generating functions, we obtain several identities involving Stirling numbers, Bernoulli numbers of higher order, generalized array polynomials, and Frobenius–Euler numbers and polynomials. We also investigate special cases of the obtained formulas and show their connections with known identities for Apostol–Bernoulli polynomials and numbers. Furthermore, exponential Euler‐type spline functions are constructed in terms of Apostol–Bernoulli numbers and polynomials. These results provide a unified approach to some classes of Bernoulli‐type and Euler‐type polynomial families.