In this paper, the notion of $\it{weighted~Toep}$-$\it{Hank}$ operator $G_{\phi}^{\beta}$, induced by the symbol $\phi\in L^{\infty}(\beta)$, on the space $H^2(\beta)$, $\beta=\{\beta_n\}_{n\in \mathbb{Z}}$ being a semi-dual sequence of positive numbers with $\beta_0=1$, is introduced. Symbols are identified for the induced $\it{weighted~Toep}$-$\it{Hank}$ operator to be co-isometry, normal and hyponormal.