On Apostol-Type Mersenne-Bernoulli and Mersenne-Euler Polynomials

In this paper, we introduce the Apostol-type Mersenne-Bernoulli and Mersenne-Euler polynomials of order $\alpha$. By employing the $M$-calculus, based on the Mersenne numbers, we establish explicit series representations, addition theorems, difference equations and convolution identities.