Let D be an (m\(\times n)\)-dimensional matrix with integer entries and let \(\mu \in {\mathbb{Z}}^ n\). Given two power series \(f(\xi)=\sum a(\alpha)\xi^{\alpha}\), \((\xi \in {\mathbb{C}}^ n\), \(\alpha \in ({\mathbb{Z}}_+)^ n)\) and \(g(\eta)=\sum b(\beta)\eta^{\beta}\) \((\eta \in {\mathbb{C}}^ m\), \(\beta \in ({\mathbb{Z}}_+)^ m)\) the author studies the generalized Hadamard composition of f and g defined as follows: \(h_{\mu}(z)=\sum a(\beta D+\mu)b(\beta)z^{\beta}\) \((z\in {\mathbb{C}}^ m\), \(\beta \in ({\mathbb{Z}}_+)^ m\), \(\beta D+\mu \geq 0)\). Some particular cases of D and \(\mu\) were previously investigated by several authors.