On the Fock Kernel for the Generalized Fock Space and Generalized Hypergeometric Series

Abstract

In this paper, we compute the reproducing kernel $B_{m,\alpha }\left(z,w\right)$ for the generalized Fock space $F_{m,\alpha }^2\left(ℂ\right)$ . 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 $B_{4,\alpha }\left(z,w\right)$ and the error function. We also obtain the closed forms of $B_{m,\alpha }\left(z,w\right)$ when $m=1,2/3,1/2$ . Finally, we also prove that $B_{m,\alpha }\left(z,z\right)~e^{\alpha {\left|z\right|}^m}{\left|z\right|}^{m-2}$ as $\mid z\mid ⟶\infty$ .