Summary: In this paper, we introduce the concept of an \(m\)-order \(n\)-dimensional generalized Hilbert tensor \(\mathcal{H}_n = (\mathcal{H}_{i_1 i_2 \cdots i_m})\), \[ \begin{aligned} \mathcal{H}_{i_1 i_2 \cdots i_m} = \frac{1}{i_1 + i_2 + \cdots i_m - m + a}, \\ a \in \mathbb{R} \setminus \mathbb{Z}^-;\;i_1, i_2, \cdots, i_m = 1, 2, \cdots, n, \end{aligned} \] and show that its \(H\)-spectral radius and its \(Z\)-spectral radius are smaller than or equal to \(M(a) n^{m - 1}\) and \(M(a) n^{\frac{m}{2}}\), respectively, here \(M(a)\) is a constant depending on \(a\). Moreover, both infinite and finite dimensional generalized Hilbert tensors are positive definite for \(a \geq 1\). For an \(m\)-order infinite dimensional generalized Hilbert tensor \(\mathcal{H}_\infty\) with \(a > 0\), we prove that \(\mathcal{H}_\infty\) defines a bounded and positively \((m - 1)\)-homogeneous operator from \(l^1\) into \(l^p\) (\(1 < p < \infty\)). Two upper bounds on the norms of corresponding positively homogeneous operators are obtained.