Padé approximation and Apostol–Bernoulli and Apostol–Euler polynomials

Using the Padé approximation of the exponential function, we obtain recurrence relations between Apostol–Bernoulli and between Apostol–Euler polynomials. As applications, we derive some new lacunary recurrence relations for Bernoulli and Euler polynomials with gap of length 4 and lacunary relations for Bernoulli and Euler numbers with gap of length 6.