Let $\mu$ be a positive Borel measure on the interval $[0,1)$. For $\alpha>0$, the generalized Hankel matrix $\mathcal{ H}_{ \mu, \alpha}=(\mu_{ n, k, \alpha})_{ n, k \geq 0}$ with entries $\mu_{ n, k, \alpha}=\int_{ [0,1)} \frac{ \Gamma(n+\alpha)}{ n ! \Gamma(\alpha)} t^{ n+k} \mathrm{ d}\mu(t)$ induces formally the operator \begin{ equation*} \mathcal{ H}_{ \mu, \alpha}(f)(z)=\sum_{ n=0}^{ \infty}\left(\sum_{ k=0}^{ \infty} \mu_{ n, k, \alpha} a_k\right) z^n \end{ equation*} on the space of all analytic function $f(z)=\sum_{ k=0}^{ \infty} a_{ k} z^{ k}$ in the unit disk $\mathbb{ D}$. In this paper, we characterize the measures $\mu$ for which $\mathcal{ H}_{ \mu, \alpha}(f)$ is well defined on the Hardy spaces $H^p(0<p<\infty)$ and satisfies $\mathcal{ H}_{ \mu, \alpha}(f)(z)=\int_{ [0,1)} \frac{ f(t)}{ (1-t z)^\alpha} \mathrm{ d} \mu(t)$. Among these measures, we further describe those for which $\mathcal{ H}_{ \mu, \alpha}(\alpha>1)$ is a bounded (resp., compact) operator from the Hardy spaces $H^p(0<p<\infty)$ into the weighted Bergman spaces $A_{ \alpha-2}^q $.
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