Abstract
In this paper, we compute the reproducing kernel B m , α z , w for the generalized Fock space F m , α 2 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 , α z , w and the error function. We also obtain the closed forms of B m , α z , w when m = 1 , 2 / 3 , 1 / 2 . Finally, we also prove that B m , α z , z ~ e α z m z m − 2 as ∣ z ∣ ⟶ ∞ .
Highlights
IntroductionThe generalized Fock kernel Bmðz, wÞ ≔ Bm,αð z, wÞ for F2mðCÞ is defined by
For any fixed parameter α > 0, we consider dλmðzÞ ≔ dλm,αðzÞ = cm,αe−αjzjm dAðzÞ, ð1Þ where dAðzÞ is the Euclidean area measure on the complex plane C
We can show that the Fock kernel can be written in terms of the Meijer-G function
Summary
The generalized Fock kernel Bmðz, wÞ ≔ Bm,αð z, wÞ for F2mðCÞ is defined by. Question: compute the Fock kernel Bmðz, wÞ for any positive rational number m. In [9], one can see the boundedness of the Bergman projection on the generalized Fock-Sobolev space with respect to dλmðzÞ. They did not obtain the explicit forms of the integral kernel. Using the properties of the incomplete gamma function, we can obtain the similar result for the generalized Fock space. We investigate the relation between GmðζÞ and generalized hypergeometric series for any positive rational number m
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