In this paper, we give relations involving values of q-Bernoulli, q-Euler, and Bernstein polynomials. Using these relations, we obtain some interesting identities on the q-Bernoulli, q-Euler, and Bernstein polynomials.

We find Fourier expansions of Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials. We give a very simple proof of them.

We introduce a special nonlinear differential operator and, using its properties, reduce higher-order Frobenius–Euler Apostol-type polynomials to a finite series of first-order Apostol-type Frobenius–Euler polynomials and Stirling numbers. Interesting identities are established.

We present explicit formulas for sums of products of Apostol-Bernoulli and Apostol-Euler numbers of the form where N and n are positive integers, B _m (q) _n stand for the Apostol-Bernoulli numbers, E _m (q) for the Apostol-Euler numbers, and \(\left( {\begin{array}{*{20}c} n \\ {m_1 , \cdots ,m_N } \\ \end{array} } \right) = \frac{{n!}} {{m_1 ! \cdots m_N !}}.\) Our formulas involve Stirling numbers of the first kind. We also derive results for Apostol-Bernoulli and Apostol-Euler polynomials. As an application, for q = 1 we recover results of Dilcher, and our paper can be regarded as a q-extension of that of Dilcher.