Small Hankel operators on generalized Fock spaces

We consider Fock spaces $F^{p,\ell}_{\alpha}$ of entire functions on ${\mathbb C}$ associated to the weights $e^{-\alpha |z|^{2\ell}}$, where $\alpha>0$ and $\ell$ is a positive integer. We compute explicitly the corresponding Bergman kernel associated to $F^{2,\ell}_{\alpha}$ and, using an adequate factorization of this kernel, we characterize the boundedness and the compactness of the small Hankel operator $\mathfrak{h}^{\ell}_{b,\alpha}$ on $F^{p,\ell}_{\alpha}$. Moreover, we also determine when $\mathfrak{h}^{\ell}_{b,\alpha}$ is a Hilbert-Schmidt operator on $F^{2,\ell}_{\alpha}$.