In this paper the main result (Theorem 2) gives the following formula for the Bernoulli polynomials \(B_ n(x)\) \[ (te^{tx}/(e^ t-1)=\sum^ \infty_{n=0}B_ n(x)t^ n/n!,\quad | t|<2\pi): \] \[ B_ n(\lambda z)=\lambda^ nB_ n(z)+n\sum^ n_{n=1}\sum^{\nu- 1}_{k=0}(-1)^ \nu{n\choose\nu}E_ \lambda(n,\nu,k)(k+\lambda z)^{n-1}, \] where \(z\) is a complex number, \(n\geq 1\) and \(\lambda\geq 2\) are integers, and \[ E_ \lambda(n,\nu,k)=\sum^{\lambda- 1}_{j=1}\varepsilon_ \lambda^{(\nu-k)j}/(1-\varepsilon^ j_ \lambda)^ n,\quad\varepsilon_ \lambda=\exp i2\pi/\lambda. \] Furthermore the author derives (Theorem 1) twelve formulas for the Bernoulli and Euler numbers and the Bernoulli and Euler polynomials, e.g. \[ B_ n=(n/2^ n(2^ n-1))\sum^ n_{\nu=1}\sum^{\nu-1}_{k=0}(- 1)^{k+1}{n\choose\nu}k^{n -1},\quad n\geq 1. \] The proofs make use of the combinatorial identity of \textit{H. W. Gould} [Combinatorial identities (1972; Zbl 0241.05011)] \[ \sum^ n_{m=k}{m-a\choose k-a}x^ m=x^ n\sum^ n_{\nu=k}{n-a+1\choose \nu-a+1}((1-x)/x)^{\nu-k} \] and the formulas of \textit{H. Alzer} [Mitt. Math. Ges. Hamb. 11, 469-471 (1987; Zbl 0632.10008)] and \textit{K. Dilcher} [Abh. Semin. Univ. Hamb. 59, 143- 156 (1989; Zbl 0712.11015)] for the Bernoulli and Euler polynomials.