  Let $\mu$ be a positive Borel measure on the interval [0,1). For $\beta > 0$,
The generalized Hankel matrix $\mathcal{H}_{\mu,\beta}=
(\mu_{n,k,\beta})_{n,k\geq0}$ with entries $\mu_{n,k,\beta}=
\int_{[0.1)}\frac{\Gamma(n+\beta)}{n!\Gamma(\beta)} t^{n+k}d\mu(t)$, induces
formally the operator $$\mathcal{H}_{\mu,\beta}(f)(z)=\sum_{n=0}^\infty
\left(\sum_{k=0}^\infty \mu_{n,k,\beta}a_k\right)z^n$$ on the space of all
analytic function $f(z)=\sum_{k=0}^ \infty a_k z^n$ in the unit disc
$\mathbb{D}$. In this paper, we characterize those positive Borel measures on
$[0,1)$ such that $\mathcal{H}_{\mu,\beta}(f)(z)= \int_{[0,1)}
\frac{f(t)}{{(1-tz)^\beta}} d\mu(t)$ for all in weighted Bergman Spaces
$A_{\alpha}^p(0<p<\infty,\; \alpha>-1)$, and among them we describe those for
which $\mathcal{H}_{\mu,\beta}(\beta>0)$ is a bounded(resp.,compact) operator
on weighted Bergman spaces and Dirichlet spaces.
