A note on the Apostol-Bernoulli and Apostol-Euler polynomials

Abstract

Let alpha is an element of N-0 = {0, 1, 2,...}. In this paper, we show provide several relationships between the generalized Apostol-Bernoulli polynomials B-n((alpha))(x; lambda) and the generalized Apostol-Euler polynomials E-n((alpha)) (x; lambda) which involve both the main results of LUO-SRIVASTAVA in [Q.-M. Luo and H. M. SRIVASTAVA, Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl. 51 (2006), 631-642] and the main results of SRIVASTAVA-PINTER in [H. M. SRIVASTAVA and A. PINTER, Remarks on some relationships between the Bernoulli and Euler polynomials, Appl. Math. Lett. 17 (4) (2004), 375-380] in the case of alpha is an element of N-0.