Explicit Formulas of the Bergman Kernel for Some Reinhardt Domains

Beberok, Tomasz

doi:10.1007/s12220-015-9652-0

Cited by 3 publications

(4 citation statements)

References 12 publications

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“…The Reinhardt domain of holomoprhy [1][2][3] has attracted the attention of many investigators in multidimensional complex analysis. This interest is generated by its geometric and analytical properties and its universality because it overlaps the balls, the polydiscs, and the Thullen domains [4] as the partial cases.…”

Section: Introduction Main Notations and Definitionsmentioning

confidence: 99%

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Analytic Functions in a Complete Reinhardt Domain Having Bounded L-Index in Joint Variables

Bandura,

Salo,

Skaskiv

2024

Symmetry

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The manuscript is an initiative to construct a full and exhaustive theory of analytical multivariate functions in any complete Reinhardt domain by introducing the concept of L-index in joint variables for these functions for a given continuous, non-negative, non-vanishing, vector-valued mapping L defined in an interior of the domain with some behavior restrictions. The complete Reinhardt domain is an example of a domain having a circular symmetry in each complex dimension. Our results are based on the results obtained for such classes of holomorphic functions: entire multivariate functions, as well as functions which are analytical in the unit ball, in the unit polydisc, and in the Cartesian product of the complex plane and the unit disc. For a full exhaustion of the domain, polydiscs with some radii and centers are used. Estimates of the maximum modulus for partial derivatives of the functions belonging to the class are presented. The maximum is evaluated at the skeleton of some polydiscs with any center and with some radii depending on the center and the function L and, at most, it equals a some constant multiplied by the partial derivative modulus at the center of the polydisc. Other obtained statements are similar to the described one.

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Section: Introduction Main Notations and Definitionsmentioning

confidence: 99%

“…Recent publications [3,10,13,14] show the growth of adaptations of one-dimensional complex analytic methods to such a domain as the Reinhardt domain in multidimensional complex analysis. Its increasing value is justified by its properties.…”

Section: Introduction Main Notations and Definitionsmentioning

confidence: 99%

Analytic Functions in a Complete Reinhardt Domain Having Bounded L-Index in Joint Variables

Bandura,

Salo,

Skaskiv

2024

Symmetry

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show abstract

“…In the theory of the Bergman kernel, it is difficult to find the closed form of the Bergman kernel for a general domain. Instead, in the case of a complex ellipsoid or similar domains, one can see the expression of the Bergman kernel in terms of the hypergeometric series in [3,4].…”

Section: Introductionmentioning

confidence: 99%

“…is closely connected with 2 F 1 and its higher dimensional hypergeometric series (Appell hypergeometric series or Lauricella hypergeometric series). Using the theory of the hypergeometric series, new formulas of the Bergman kernel have been computed in [3,[5][6][7].…”

Section: Introductionmentioning

confidence: 99%

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

Park

2021

Journal of Function Spaces

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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 ∣ ⟶ ∞ .

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Equivalence of invariant metrics via Bergman kernel on complete noncompact Kähler manifolds

Cho,

Lee

2023

Math. Ann.

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Explicit Formulas of the Bergman Kernel for Some Reinhardt Domains

Abstract: In this paper we obtain the closed forms of some hypergeometric functions. As an application, we obtain the explicit forms of the Bergman kernel functions for Reinhardt domains

Cited by 3 publications

References 12 publications

Analytic Functions in a Complete Reinhardt Domain Having Bounded L-Index in Joint Variables

Analytic Functions in a Complete Reinhardt Domain Having Bounded L-Index in Joint Variables

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

Equivalence of invariant metrics via Bergman kernel on complete noncompact Kähler manifolds

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