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

Abstract

In this paper, we compute the reproducing kernel Bm,αz,w for the generalized Fock space Fm,α2ℂ. The usual Fock space is the case when m=2. We express the reproducing kernel in terms of a suitable hypergeometric series 1Fq. In particular, we show that there is a close connection between B4,αz,w and the error function. We also obtain the closed forms of Bm,αz,w when m=1,2/3,1/2. Finally, we also prove that Bm,αz,z~eαzmzm−2 as ∣z∣⟶∞.