Small Hankel operators on generalized Fock spaces

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