On the Fock kernel for the generalized Fock space and generalized hypergeometric series (Q2230005)

Summary: In this paper, we compute the reproducing kernel \(B_{m, \alpha}(z, w)\) for the generalized Fock space \(F_{m, \alpha}^2(\mathbb{C})\). The usual Fock space is the case when \(m=2\). We express the reproducing kernel in terms of a suitable hypergeometric series \({}_1 F_q\). In particular, we show that there is a close connection between \(B_{4, \alpha}(z, w)\) and the error function. We also obtain the closed forms of \(B_{m, \alpha}(z, w)\) when \(m=1,2/3,1/2\). Finally, we also prove that \(B_{m, \alpha}(z, z)\sim e^{\alpha |z|^m} |z|^{m-2}\) as \(|z|\longrightarrow\infty\).