The multiplication formulas for the Apostol–Bernoulli and Apostol–Euler polynomials of higher order

This article obtains the multiplication formulas for the Apostol–Bernoulli and Apostol–Euler polynomials of higher order and deduces some explicit recursive formulas. The λ-multiple power sum and λ-multiple alternating sum that are clearly evaluated associated with the Apostol–Bernoulli and Apostol–Euler polynomials of higher order, respectively, are also introduced. Some earlier results of Carlitz and Howard can be deduced.