Transformation formulas for the Bergman kernels and projections of Reinhardt domains

Let $\mathbb{Z}$ and $\mathbb{Z}^{+}$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in \mathbb{Z}^{+}$, let $t(a,b,c,d;n)$ be the number of representations of $n$ by $\frac{1}{2}ax(x+1)+\frac{1}{2}by(y+1)+\frac{1}{2}cz(z+1)+\frac{1}{2}dw(w+1)$ with $x,y,z,w\in \mathbb{Z}$. Using theta function identities we prove 13 transformation formulas for $t(a,b,c,d;n)$ and evaluate $t(2,3,3,8;n)$, $t(1,1,6,24;n)$ and $t(1,1,6,8;n)$.

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Generalized reinhardt domains

Journal of Geometric Analysis ◽

10.1007/bf02938112 ◽

1991 ◽

Vol 1 (1) ◽

pp. 1-17 ◽

Cited By ~ 1

Author(s):

Eric Bedford ◽

Jiri Dadok

Keyword(s):

Reinhardt Domains

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Positivity of the Poisson Kernel for the Continuous q-Jacobi Polynomials and Some Quadratic Transformation Formulas for Basic Hypergeometric Series

SIAM Journal on Mathematical Analysis ◽

10.1137/0517069 ◽

1986 ◽

Vol 17 (4) ◽

pp. 970-999 ◽

Cited By ~ 11

Author(s):

George Gasper ◽

Mizan Rahman

Keyword(s):

Hypergeometric Series ◽

Poisson Kernel ◽

Jacobi Polynomials ◽

Basic Hypergeometric Series ◽

Quadratic Transformation ◽

Transformation Formulas

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BERGMAN KERNELS AND EQUILIBRIUM MEASURES FOR POLARIZED PSEUDO-CONCAVE DOMAINS

International Journal of Mathematics ◽

10.1142/s0129167x10005933 ◽

2010 ◽

Vol 21 (01) ◽

pp. 77-115 ◽

Cited By ~ 2

Author(s):

ROBERT J. BERMAN

Keyword(s):

Toeplitz Operators ◽

Equilibrium Measure ◽

Markov Property ◽

General Setting ◽

Space Structure ◽

Tensor Power ◽

Bergman Kernels ◽

Equilibrium Measures ◽

Holomorphic Sections ◽

Positive Line Bundle

Let X be a domain in a closed polarized complex manifold (Y,L), where L is a (semi-) positive line bundle over Y. Any given Hermitian metric on L induces by restriction to X a Hilbert space structure on the space of global holomorphic sections on Y with values in the k-th tensor power of L (also using a volume form ωn on X. In this paper the leading large k asymptotics for the corresponding Bergman kernels and metrics are obtained in the case when X is a pseudo-concave domain with smooth boundary (under a certain compatibility assumption). The asymptotics are expressed in terms of the curvature of L and the boundary of X. The convergence of the Bergman metrics is obtained in a more general setting where (X,ωn) is replaced by any measure satisfying a Bernstein–Markov property. As an application the (generalized) equilibrium measure of the polarized pseudo-concave domain X is computed explicitly. Applications to the zero and mass distribution of random holomorphic sections and the eigenvalue distribution of Toeplitz operators will be described elsewhere.

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Mapping properties of the Bergman projections on elementary Reinhardt domains

Mathematica Slovaca ◽

10.1515/ms-2021-0024 ◽

2021 ◽

Vol 71 (4) ◽

pp. 831-844

Author(s):

Shuo Zhang

Keyword(s):

Sobolev Spaces ◽

Bergman Projection ◽

Reinhardt Domain ◽

Reinhardt Domains ◽

Multi Index ◽

Bergman Projections

Abstract The elementary Reinhardt domain associated to multi-index k = (k 1, …, k n ) ∈ ℤ n is defined by ℋ ( k ) : = { z ∈ D n : z k is defined and | z k | < 1 } . $$\mathcal{H}(\mathbf{k}):=\{z\in\mathbb{D}^n: z^{\mathbf{k}}\ \text{is defined and}\ |z^{\mathbf{k}}|<1\}.$$ In this paper, we study the mapping properties of the associated Bergman projection on L p spaces and L p Sobolev spaces of order ≥ 1.

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