基于二重加密的数字水印算法

占履军; 李昇; 张旭东

doi:10.6054/j.jscnun.2017156

摘要: 为了保护两人或者多人共有的版权利益不因单个人的泄密受到影响，本文提出了用Arnold置乱和采样置乱对水印图像进行了二重加密的方法，达到了共有图像版权保护的目的。论文还通过对二重加密与单次加密作比较，表明二重加密比单次加密的鲁棒性更好，但不可见性略差。

关键词:

采样置乱

Abstract: In order to guarantee the copyright of two or more people and protect it from single leak, this paper introduces the Arnold scrambling and the sample scrambling for watermark double encryption. The invisibility and robustness are observed and compared with single scrambling. The experimental results proved that the robustness of the double encryption is better than that of single encryption with a slightly worse invisibility. It can achieve better confidentiality requirements of copyright protection.

Keywords:

Sample scrambling

1958年，华罗庚^[1]得到了4类Cartan域的Bergman核函数的显式表达；1999、2000年，殷慰萍^[2-3]得到了4类超Cartan域的Bergman核函数的显式表达；2001年，殷慰萍等^[4-7]引入了4类Cartan-Egg域，并将Cartan-Egg域进一步推广为华罗庚域。

华罗庚域及相关问题已有许多研究结果。如：苏简兵等^[8]研究了广义华罗庚域的Bergman核函数的计算方法；殷慰萍^[9]综述了华罗庚域的Einstein-Kähler度量的计算方法；仲小丽等^[10]研究了华罗庚域的凸性并计算了第一类华罗庚域的Kobayashi度量；尹明和刘名生^[11]得到了某些华罗庚域的Bergman核函数的显式表达；李海涛等^[12]讨论了第二类华罗庚域的极值问题；ZHAO和WANG^[13]研究了一类华罗庚域的陆启铿问题；程晓亮和马会波^[14]研究了一类华罗庚域与复欧氏空间的不相关性。

本文将在任意不可约有界圆型齐性域上考虑下面类型的域：

$E\left(q_{1}, \cdots, q_{m}, {{\mathit{\Omega}}} ; p_{1}, \cdots, p_{m}\right):=\left\{\left(W_{1}, \cdots, W_{m}, Z\right) \in\right. \\ \quad \mathbb{C}^{q_{1}} \times \cdots \times \mathbb{C}^{q_{m}} \times {{\mathit{\Omega}}} ;\left\|W_{1}\right\|^{2 p_{1}}+\left\|W_{2}\right\|^{2 p_{2}}+\cdots+\left\|W_{m}\right\|^{2 p_{m}}< \\ \quad N(Z, Z)\}, \\$

记E(q₁, …, q_m, Ω; p₁, …, p_m)为E_Ω，其中Ω是指任意不可约有界圆型齐性域，q₁, …, q_m都是自然数，m, p₁, …, p_m都是正整数，N(Z, Z)是Ω的一般范数。当p₁, …, p_m都是正整数时，域E_Ω的Bergman核函数可由多维超几何函数表达，但并非显式表达。域E_Ω将华罗庚域的底域推广至任意不可约有界圆型齐性域，当Ω为4类典型域时，域E_Ω就是华罗庚域。本文主要研究了当1/p₁, …, 1/p_m－1是正整数，p_m < 1是正实数时，域E_Ω的Bergman核函数的显式表达。

1. 预备知识

下面给出本文主要结论证明中需要用到的几个引理。

引理1^[4] 假定D是 $\mathbb{C}^{m+n}$ 中的semi-Reinhardt域，则D的完备正交系为：

$\left\{w_{1}^{j_{1}} \cdots w_{m}^{j_{m}} P_{k i}^{(j)}(z) ; j=\left(j_{1}, \cdots, j_{m}\right) \in \mathbb{N}^{m}, k \in \mathbb{N}, 1 \leqslant i \leqslant m_{k}\right\},$

其中, m_k=(n+k－1)! [k!(n－1)!]^－1。

引理2^[4] 假设P(x)是x的n次多项式P(x)=a_nxⁿ+a_n－1x^n－1+…+a₁x+a₀，则P(x)可以改写为如下形式：

$P(x)= b_{n}(x+n) \cdots(x+1)+b_{n-1}(x+n-1) \cdots(x+1)+ \\ \;\;\;\;\;\;\;\; \cdots+b_{1}(x+1)+b_{0}=\sum\limits_{j=0}^{n} \frac{b_{j} \Gamma(x+j+1)}{\Gamma(x+1)},$

其中, Γ是伽马函数。

引理3^[4] 当|t| < 1，s>0时，有

$\sum\limits_{k=0}^{\infty} \frac{\Gamma(k+s)}{\Gamma(s) \Gamma(k+1)} t^{k}=(1-t)^{-s} 。$

引理4^[5] 假设

$\begin{aligned} & h_{s}\left(x_{1}, \cdots, x_{N-1}\right)= \\ & \quad \sum\limits_{j_{1}, \cdots, j_{N-1}=0}^{\infty} \frac{\Gamma\left(s+\sum\nolimits_{l=1}^{N-1}\left(\left(j_{l}+1\right) / p_{l}\right)\right)}{\prod\nolimits_{l=1}^{N-1} \Gamma\left(\left(j_{l}+1\right) / p_{l}\right)} x_{1}^{j_{1}} x_{2}^{j_{2}} \cdots x_{N-1}^{j_{N-1}}, \end{aligned}$

其中, s>0，1/p_l为正整数，1≤l≤N－1， $\sum\limits_{l=1}^{N-1}\left|x_{l}\right|^{p_{l}} < 1$ ，则

$h_{s}\left(x_{1}, \cdots, x_{N-1}\right)= \\ \;\;\;\;\;\; \quad \frac{\partial^{N-1}}{\partial x_{1} \cdots \partial x_{N-1}}\left(\sum\limits_{k_{1}=0}^{q_{1}-1} \cdots \sum\limits_{k_{N-1}=0}^{q_{N-1}-1} \frac{\Gamma(s)}{\left(1-\sum\nolimits_{l=1}^{N-1} \omega_{l}^{k_{l}} x_{l}^{1 / q_{l}}\right)^{s}}\right),$

其中，q_l=1/p_l，ω_l=e^2πi/_{q_l}，x_l^1/_{q_l}=|x_l|^1/q_le^φ_li/q_l，－π < φ_l=arc x_l < π，1≤l≤N－1。

引理5^[15] 假定复向量空间V为Jordan三元系统，Ω是V中任意不可约有界圆型齐性域，ω=αⁿ是V上的体积形式。相对于αⁿ，设V_Ω是Ω的体积，对s∈ $\mathbb{C}$ ，Re s>－1，有

$\int_{{\mathit{\Omega}}} N(Z, Z)^{s} \alpha^{n}=\frac{\chi(0)}{\chi(s)} V_{{\mathit{\Omega}}},$

其中，N(Z, Z)是Ω的一般范数，χ(s)= $\prod\limits_{j=1}^r\left(s+1+(j-1) \frac{a}{2}\right)_{1+b+(r-j) a}$ ，这里a=dim V_ij (0 < i < j)，b=dim V_0i (0 < i) 是V的数字不变量，r是V的秩。

引理6 设Ω的亏格为g，Φ∈(Aut Ω)₀为Ω的全纯自同构群的单位分支，JΦ(Z)=det DΦ(Z)为Φ在Z∈Ω的复雅可比，则对任意Φ∈(Aut Ω)₀及模为1的复数α₁, α₂, …, α_m，由

${\mathit{\Psi}}\left(W_{1}, \cdots, W_{m}, Z\right)= \\ \;\;\;\;\;\; \quad\left(\alpha_{1} A\left(Z, Z_{0}\right)^{1 / p_{1}} W_{1}, \cdots, \alpha_{m} A\left(Z, Z_{0}\right)^{1 / p_{m}} W_{m}, {\mathit{\Phi}}(Z)\right)$

定义的映射Ψ: E(Ω; p₁, …, p_m)↦E(Ω; p₁, …, p_m)是E(Ω; p₁, …, p_m)=E(1, …, 1, Ω; p₁, …, p_m)的全纯自同构，其中A(Z, Z₀)=N(Z₀, Z₀)^1/2/N(Z, Z₀)，Z₀=Φ^－1(0)，可得

$\left|J {\mathit{\Psi}}\left(W_{1}, \cdots, W_{m}, Z_{0}\right)\right|^{2}=N\left(Z_{0}, Z_{0}\right)^{-\sum\nolimits_{i=1}^{m}{ }^{1 / p_{i}-g}}。$

类似文献[15]中引理3.1的证明方法，易证引理6成立，在此略。

2. 主要结论及其证明

考虑q₁=…=q_m=1，简记E(Ω; p₁, …, p_m)=E(1, …, 1, Ω; p₁, …, p_m)。以K_E表示E(Ω; p₁, …, p_m)的Bergman核函数，Φ∈(Aut Ω)₀是Ω的自同构映射，并将Z∈Ω映为0，记W₁^*=A(Z, Z)^1/p₁W₁, …, W_m^*=A(Z, Z)^1/p_mW_m。

根据Bergman核的变换法则及|JΨ(W₁, …, W_m, Z)|²= $N(Z, Z)^{-\sum_{i=1}^m 1 / p_i-g}$ ，有

$K_{E}\left(W_{1}, \cdots, W_{m}, Z\right)= \\ \;\;\;\;\;\; K_{E}\left(W_{1}^{*}, \cdots, W_{m}^{*}, 0\right) N(Z, Z)^{-\sum\nolimits_{i=1}^{m} 1_{p_{i}-g}},$

故K_E(W₁, …, W_m, Z)的确定转化为计算K_E(W₁^*, …, W_m^*, 0)。

E(Ω; p₁, …, p_m)显然是 $\mathbb{C}^{m+n}$ 中的semi-Reinhardt域，由引理1知，其上有完备正交系：

$\begin{gathered} \left\{W_{1}^{j_{1}} W_{2}^{j_{2}} \cdots W_{m}^{j_{m}} P_{l i}^{\left(j_{1}, \cdots, j_{m}\right)}(Z) ; j_{1}, \cdots, j_{m}, l \in \mathbb{N}\right., \\ \left.1 \leqslant i \leqslant m_{l}=\binom{n+l-1}{l}\right\}, \end{gathered}$

其中，P_li^{(j₁, …, j_m)}是l次多项式，l∈ $\mathbb{N}$ 。特别地，P_0i^{(j₁, …, j_m)}是常数多项式，a_{j₁, …, j_m}由下式定义^[11]：

$\begin{gathered} \left|a_{j_{1}, \cdots, j_{m}}\right|^{2} \int_{E\left({\mathit{\Omega}} ; p_{1}, \cdots, p_{m}\right)}\left|W_{1}^{j_{1}} W_{2}^{j_{2}} \cdots W_{m}^{j_{m}}\right|^{2} \omega_{1}\left(W_{1}\right) \wedge \cdots \wedge \\ \omega_{m}\left(W_{m}\right) \wedge \omega(Z)=1 。 \end{gathered}$

于是E(Ω; p₁, …, p_m)在(W₁^*, …, W_m^*, 0)的Bergman核为：

$K_{E}\left(W_{1}^{*}, \cdots, W_{m}^{*}, 0\right)=\sum\left|W_{1}^{* j_{1}} \cdots W_{m}^{* j_{m}} P_{l i}^{\left(j_{1}, \cdots, j_{m}\right)}(0)\right|^{2}= \\ \;\;\;\;\;\; \sum\left|a_{j_{1}, \cdots, j_{m}}\right|^{2}\left|W_{1}^{* j_{1}}\right|^{2} \cdots\left|W_{m}^{* j_{m}}\right|^{2} 。$

下面先给出一个引理。

引理7 设(W₁, W₂, …, W_m)∈ $\mathbb{C}^{m}$ 满足|W₁|^2p₁+|W₂|^2p₂+…+|W_m|^2p_m < R²时，这里R>0，则

$\begin{aligned} & \int_{\left|W_{1}\right|^{2p_{1}}+\left|W_{2}\right|^{2p_{2}}+\cdots+\left|W_{m}\right|^{2p_{m}}<R^{2}}\left[\left|W_{1}\right|^{2 j_{1}}\left|W_{2}\right|^{2 j_{2}} \cdots\left|W_{m}\right|^{2 j_{m}} \times\right. \\ &\;\;\;\;\;\; \left.\omega\left(W_{1}\right) \wedge \cdots \wedge \omega\left(W_{m}\right)\right]=\frac{1}{p_{1} \cdots p_{m}} \times \\ &\;\;\;\;\;\; \frac{\Gamma\left(\left(j_{1}+1\right) / p_{1}\right) \cdots \Gamma\left(\left(j_{m}+1\right) / p_{m}\right)}{\Gamma\left(\sum\nolimits_{i=1}^{m}\left(\left(j_{i}+1\right) / p_{i}\right)+1\right)} R^{2 \sum\nolimits_{i=1}^{m}\left(\left(j_{i}+1\right) / p_{i}\right)}。 \end{aligned}$

证明令W_i=r_ie^iθ_i，s_i=r_i^p_i，其中r_i≥0，s_i≥0，1≤i≤m，则

$\int {_{{{\left| {{W_1}} \right|}^{2p_1}} + {{\left| {{W_2}} \right|}^{2p_2}} + \cdots + {{\left| {{W_m}} \right|}^{2p_m}} < {R^2}}\left[ {{{\left| {{W_1}} \right|}^{2{j_1}}}{{\left| {{W_2}} \right|}^{2{j_2}}} \cdots {{\left| {{W_m}} \right|}^{2{j_m}}}} \right. \times } \\ \;\;\;\;\;\; \left. {\omega \left( {{W_1}} \right) \wedge \cdots \wedge \omega \left( {{W_m}} \right)} \right] = \\ \;\;\;\;\;\; {2^m}\int {_{r_1^{2{p_1}} + r_2^{2{p_2}} + \cdots + r_m^{2{p_m}} < {R^2}}r_1^{2{j_1} + 1} \cdots r_m^{2{j_m} + 1}} {\rm{d}}{\mathit{r}_1} \cdots {\rm{d}}{\mathit{r}_m} = \\ \;\;\;\;\;\; \frac{{{2^m}}}{{{p_1} \cdots {p_m}}}\int {_{s_1^2 + \cdots + s_m^2 < {R^2}}} s_1^{\left( {2{j_1} + 2} \right)/{p_1} - 1} \cdots s_m^{\left( {2{j_m} + 2} \right)/{p_m} - 1}{\rm{d}}{\mathit{s}_1} \cdots {\rm{d}}{\mathit{s}_m}。$

应用多元极坐标变换，可得

$\int_{s_{1}^{2}+\cdots+s_{m}^{2}<R^{2}} s_{1}^{\left(2 j_{1}+2\right) / p_{1}-1} \cdots s_{m}^{\left(2 j_{m}+2\right) / p_{m}-1} \mathrm{d} s_{1} \cdots \mathrm{d} s_{m}=\\ \;\;\;\;\;\; \int_{0}^{R} \rho^{\left(2 j_{1}+2\right) / p_{1}+\cdots+\left(2 j_{m}+2\right) / p_{m}-1} \mathrm{d} \rho \times\\ \;\;\;\;\;\; \int_{0}^{{\mathsf{π}} / 2} \cdots \int_{0}^{{\mathsf{π}} / 2}\left[\left(\left(\cos \theta_{1}\right)^{\left(2 j_{1}+2\right) / p_{1}-1}\left(\sin \theta_{1} \cos \theta_{2}\right)^{\left(2 j_{2}+2\right) / p_{2}-1} \times\right.\right.\\ \;\;\;\;\;\; \cdots \times\left(\sin \theta_{1} \cdots \sin \theta_{m-1}\right)^{\left(2 j_{m}+2\right) / p_{m}-1}\left(\sin \theta_{1}\right)^{m-2}\left(\sin \theta_{2}\right)^{m-3} \times\\ \;\;\;\;\;\; \left.\left.\cdots \times\left(\sin \theta_{m-2}\right)\right)\right] \mathrm{d} \theta_{1} \cdots \mathrm{d} \theta_{m-1}=\\ \;\;\;\;\;\; \int_{0}^{R} \rho^{\left(2 j_{1}+2\right) / p_{1}+\cdots+\left(2 j_{m}+2\right) / p_{m}-1} \mathrm{d} \rho \int_{0}^{{\mathsf{π}} / 2}\left[\left(\cos \theta_{1}\right)^{\left(2 j_{1}+2\right) / p_{1}-1} \times\right.\\ \;\;\;\;\;\; \left.\left(\sin \theta_{1}\right)^{\left(2 j_{2}+2\right) / p_{2}-1+\cdots+\left(2 j_{m}+2\right) / p_{m}-1+m-2}\right] \mathrm{d} \theta_{1} \times \cdots \times\\ \;\;\;\;\;\; \int_{0}^{{\mathsf{π}} / 2}\left(\cos \theta_{m-1}\right)^{\left(2 j_{m-1}+2\right) / p_{m-1}-1}\left(\sin \theta_{m-1}\right)^{\left(2 j_{m}+2\right) / p_{m}-1} \mathrm{d} \theta_{m-1}=\\ \;\;\;\;\;\; \frac{R^{\sum\nolimits_{i=1}^{m}\left(\left(2 j_{i}+2\right) / p_{i}\right)}}{\sum\nolimits_{i=1}^{m}\left(\left(2 j_{i}+2\right) / p_{i}\right)} \times\\ \;\;\;\;\;\; \frac{\Gamma\left(\left(j_{2}+1\right) / p_{2}+\cdots+\left(j_{m}+1\right) / p_{m}\right) \Gamma\left(\left(j_{1}+1\right) / p_{1}\right)}{2 \Gamma\left(\left(j_{1}+1\right) / p_{1}+\cdots+\left(j_{m}+1\right) / p_{m}\right)} \times \cdots \times\\ \;\;\;\;\;\; \frac{\Gamma\left(\left(j_{m}+1\right) / p_{m}\right) \Gamma\left(\left(j_{m-1}+1\right) / p_{m-1}\right)}{2 \Gamma\left(\left(j_{m-1}+1\right) / p_{m-1}+\left(j_{m}+1\right) / p_{m}\right)},$

从而可得

$\int_{\left|W_1\right|^{2p_1}+\left|W_2\right|^{2p_2}+\cdots+\left|W_m\right|^{2p_m}<R^2}\left[\left|W_1\right|^{2 j_1}\left|W_2\right|^{2 j_2} \cdots\left|W_m\right|^{2 j_m} \times\right.\\ \;\;\; \left.\omega\left(W_{1}\right) \wedge \cdots \wedge \omega\left(W_{m}\right)\right]= \\ \;\;\; \frac{2^{m}}{p_{1} \cdots p_{m}} \frac{R^{\sum\nolimits_{i=1}^{m}\left(\left(2 j_{i}+2\right) / p_{i}\right)}}{\sum\nolimits_{i=1}^{m}\left(\left(2 j_{i}+2\right) / p_{i}\right)} \times \\ \;\;\; \frac{\Gamma\left(\left(j_{1}+1\right) / p_{1}\right) \cdots \Gamma\left(\left(j_{m}+1\right) / p_{m}\right)}{2^{m-1} \Gamma\left(\sum\nolimits_{i=1}^{m}\left(j_{i}+1\right) / p_{i}\right)}=\frac{1}{p_{1} \cdots p_{m}} \times \\ \;\;\; \frac{\Gamma\left(\left(j_{1}+1\right) / p_{1}\right) \cdots \Gamma\left(\left(j_{m}+1\right) / p_{m}\right)}{\Gamma\left(\sum\nolimits_{i=1}^{m}\left(\left(j_{i}+1\right) / p_{i}\right)+1\right)} R^{2 \sum\nolimits_{i=1}^{m}\left(\left(j_{i}+1\right) / p_{i}\right)}。$

证毕。

由引理7可得到E(Ω; p₁, …, p_m)的Bergman核函数：

定理1 E(Ω; p₁, …, p_m)的Bergman核函数为

$K_E\left(W_1, \cdots, W_m, Z\right)=\\ \;\;\;\;\;\; {\mathit{\Lambda}}\left(\frac{\left|W_{1}\right|^{2}}{N(Z, Z)^{1 / p_{1}}}, \cdots, \frac{\left|W_{m}\right|^{2}}{N(Z, Z)^{1 / p_{m}}}\right) N(Z, Z)^{-\sum\nolimits_{i=1}^{m} 1^{1 / p_{i}-g}},$

其中,

${\mathit{\Lambda}}\left(t_{1}, \cdots, t_{m}\right)=\frac{p_{1} \cdots p_{m}}{\chi(0) V_{{\mathit{\Omega}}}} \frac{\partial^{m-1}}{\partial t_{1} \cdots \partial t_{m-1}} \times \\ \;\;\;\;\;\; \sum\limits_{k_{1}=0}^{r_{1}-1} \cdots \sum\limits_{k_{m-1}=0}^{r_{m-1}{-1}} \sum\limits_{j=1}^{n+2} \sum\limits_{u=0}^{j} \sum\limits_{v=0}^{u-1} \frac{b_{j} b_{u}(j) b_{v}(u) \Gamma(v+1)}{\left(1-\sum\limits_{i=1}^{m-1} \omega_{i}^{k_{i}} t_{i}^{1 / r_{i}}\right)^{1 / p_{m}+u-1}(1-x)^{v+1}}。$

证明首先计算|a_{j₁, …, j_m}|^－2。由a_{j₁, …, j_m}定义，可得到

$\left|a_{j_{1}, \cdots, j_{m}}\right|^{-2}=\\ \;\;\; \int_{Z \in {\mathit{\Omega}}}\left[\int_{\left|W_{1}\right| 2 p_{1}+\left|W_{2}\right| 2 p_{2}+\cdots+\left|W_{m}\right| 2 p_{m}<N(Z, Z)}\left(\left|W_{1}^{j_{1}} W_{2}^{j_{2}} \cdots W_{m}^{j_{m}}\right|^{2} \times\right.\right.\\ \;\;\; \left.\left.\omega_{1}\left(W_{1}\right) \wedge \cdots \wedge \omega_{m}\left(W_{m}\right)\right)\right] \wedge \omega(Z)。$

令 $h=\sum\limits_{i=1}^{m} \frac{j_{i}+1}{p_{i}}$ ，N(Z, Z)=R²，则由引理5和引理7可得

$\left|a_{j_{1}, \cdots, j_{m}}\right|^{2}=\frac{p_{1} \cdots p_{m} \chi(h) \Gamma(h+1)}{\chi(0) V_{{\mathit{\Omega}}} \Gamma\left(\left(j_{1}+1\right) / p_{1}\right) \cdots \Gamma\left(\left(j_{m}+1\right) / p_{m}\right)}。$

由于

$h(h-1) \chi(h)=\sum\limits_{j=1}^{n+2} b_{j}(h+1)_{j}=\sum\limits_{j=1}^{n+2} b_{j} \Gamma(h+j+1) / \Gamma(h+1),$

其中，b_j(1≤j≤n+2)为常数。进而有

$\left|a_{j_1, \cdots, j_m}\right|^2=\\ \;\;\;\;\;\; \frac{p_{1} \cdots p_{m} \Gamma(h-1) \sum\nolimits_{j=1}^{n+2} b_{j} \Gamma(h+j+1) / \Gamma(h+1)}{\chi(0) V_{{\mathit{\Omega}}} \Gamma\left(\left(j_{1}+1\right) / p_{1}\right) \cdots \Gamma\left(\left(j_{m}+1\right) / p_{m}\right)}= \\ \;\;\;\;\;\; \frac{p_{1} \cdots p_{m}}{\chi(0) V_{{\mathit{\Omega}}}} \sum\limits_{j=1}^{n+2} b_{j} \frac{\Gamma(h+j+1) \Gamma(h-1)}{\Gamma\left(\left(j_{1}+1\right) / p_{1}\right) \cdots \Gamma\left(\left(j_{m}+1\right) / p_{m}\right) \Gamma(h+1)},$

从而可得

$\begin{aligned} & K_{E}\left(W_{1}^{*}, \cdots, W_{m}^{*}, 0\right)=\frac{p_{1} \cdots p_{m}}{\chi(0) V_{{\mathit{\Omega}}}} \times \\ & \sum\limits_{j_{1}, \cdots, j_{m}=0}^{\infty} \sum\limits_{j=1}^{n+2}\left[b_{j} \frac{\Gamma(h+j+1) \Gamma(h-1)}{\Gamma\left(\left(j_{1}+1\right) / p_{1}\right) \cdots \Gamma\left(\left(j_{m}+1\right) / p_{m}\right) \Gamma(h+1)} \times\right. \\ & \left.\left|W_{1}^{* j_{1}}\right|^{2} \cdots\left|W_{m}^{* j_{m}}\right|^{2}\right] 。 \end{aligned}$

由于 $\frac{\Gamma(h+j+1)}{\Gamma(h+1)}$ =(h－2+3)…(h－2+j+2)∶=P₁(h－2)是h－2的j次多项式，由引理2知存在j+1个常数b_u(j) (u=0, 1, …, j)，使得

$P_{1}(h-2)=\sum\limits_{u=0}^{j} b_{u}(j) \frac{\Gamma(h-2+u+1)}{\Gamma(h-2+1)}。$

因此，

$K_{E}\left(W_{1}^{*}, \cdots, W_{m}^{*}, 0\right)=\frac{p_{1} \cdots p_{m}}{\chi(0) V_{{\mathit{\Omega}}}} \times \\ \;\;\;\;\;\; \sum\limits_{j_{1}, \cdots, j_{m}=0}^{\infty} \sum\limits_{j=1}^{n+2}\left[b_{j} \frac{\sum\nolimits_{u=0}^{j} b_{u}(j) \Gamma(h-1+u)}{\Gamma\left(\left(j_{1}+1\right) / p_{1}\right) \cdots \Gamma\left(\left(j_{m}+1\right) / p_{m}\right)} \times\right. \\ \;\;\;\;\;\; \left.\left|W_{1}^{*}\right|^{2 j_{1}} \cdots \mid W_{m}^{*}|^{2j_{m}}\right]=\frac{p_{1} \cdots p_{m}}{\chi(0) V_{{\mathit{\Omega}}}} \times \\ \;\;\;\;\;\; \sum\limits_{j=1}^{n+2} b_{j} \sum\limits_{u=0}^{j} b_{u}(j) \sum\limits_{j_{m}=0}^{\infty} \frac{\left|W_{m}^{*}\right|^{2j_{m}}}{\Gamma\left(\left(j_{m}+1\right) / p_{m}\right)} \times \\ \;\;\;\;\;\; \sum\limits_{j_{1}, \cdots, j_{m-1}=0}^{\infty}\left[\frac{\Gamma\left(\sum\limits_{i=1}^{m-1}\left(\left(j_{i}+1\right) / p_{i}\right)+\left(j_{m}+1\right) / p_{m}+u-1\right)}{\Gamma\left(\left(j_{1}+1\right) / p_{1}\right) \cdots \Gamma\left(\left(j_{m-1}+1\right) / p_{m-1}\right)} \times\right. \\ \;\;\;\;\;\; \left.\left|W_{1}^{*}\right|^{2j_{1}} \cdots \mid W_{m-1}^{*}\mid^{2 j_{m-1}}\right] 。$

应用引理4，取λ=(j_m+1)/p_m+u－1，t_i=|W_i^*|²(1≤i≤m)，记K_E(W₁^*, …, W_m^*, 0)=Λ(t₁, …, t_m)。显然，λ >0，并且在Z=0时，满足 $\sum\limits_{i=1}^{m-1}\left|t_i\right|^{p_i} < 1$ ，因此，

${\mathit{\Lambda}}\left(t_{1}, \cdots, t_{m}\right)=\frac{p_{1} \cdots p_{m}}{\chi(0) V_{{\mathit{\Omega}}}} \sum\limits_{j=1}^{n+2} b_{j} \sum\limits_{u=0}^{j} b_{u}(j) \times\\ \;\;\;\;\;\; \sum\limits_{j_{m}=0}^{\infty} h_{\lambda}\left(t_{1}, \cdots, t_{m-1}\right) \frac{t_{m}^{j_{m}}}{\Gamma\left(\left(j_{m}+1\right) / p_{m}\right)}=\frac{p_{1} \cdots p_{m}}{\chi(0) V_{{\mathit{\Omega}}}} \sum\limits_{j=1}^{n+2} b_{j} \sum\limits_{u=0}^{j} b_{u}(j) \times\\ \;\;\;\;\;\; \frac{\partial^{m-1}}{\partial t_{1} \cdots \partial t_{m-1}} \sum\limits_{k_{1}=0}^{r_{1}-1} \cdots \sum\limits_{k_{m-1}=0}^{r_{m-1}-1} \sum\limits_{j_{m}=0}^{\infty}\left[\frac{\Gamma\left(\left(j_{m}+1\right) / p_{m}+u-1\right)}{\Gamma\left(\left(j_{m}+1\right) / p_{m}\right)} \times\right.\\ \;\;\;\;\;\; \left.\frac{t_{m}^{j_{m}}}{\left(1-\sum\nolimits_{i=1}^{m-1} \omega_{i}^{k_{i}} t_{i}^{1 / r_{i}}\right)^{\lambda}}\right],$

其中, r_i=1/p_i∈ $\mathbb{Z}_+$ (1≤i≤m－1)。

由引理2，有

$\begin{aligned} & \frac{\Gamma\left(\left(j_{m}+1\right) / p_{m}+u-1\right)}{\Gamma\left(\left(j_{m}+1\right) / p_{m}\right)}= \\ & \quad\left(\frac{j_{m}+1}{p_{m}}+u-2\right)\left(\frac{j_{m}+1}{p_{m}}+u-3\right) \cdots\left(\frac{j_{m}+1}{p_{m}}+1\right)\left(\frac{j_{m}+1}{p_{m}}\right):= \\ & \quad P_{2}\left(j_{m}\right) \end{aligned}$

为j_m的u－1次多项式，则存在u个常数b_v(u) (v=0, 1, …, u－1)，使得

$P_{2}\left(j_{m}\right)=\sum\limits_{v=0}^{u-1} b_{v}(u) \frac{\Gamma\left(j_{m}+v+1\right)}{\Gamma\left(j_{m}+1\right)}。$

令x=t_m/ $\left(1-\sum\limits_{i=1}^{m-1} \omega_i^{k_i} t_i^{1 / r_i}\right)^{1 / p_m}$ ，则

$\sum\limits_{j_{m}=0}^{\infty} \frac{\Gamma\left(\left(j_{m}+1\right) / p_{m}+u-1\right)}{\Gamma\left(\left(j_{m}+1\right) / p_{m}\right)} \times \frac{t_{m}^{j_{m}}}{\left(1-\sum\limits_{i=1}^{m-1} \omega_{i}^{k_{i}} t_{i}^{1 / r_{i}}\right)^{\lambda}}=\\ \sum\limits_{j_{m}=0}^{\infty} \sum\limits_{v=0}^{u-1} b_{v}(u) \frac{\Gamma\left(j_{m}+v+1\right)}{\Gamma\left(j_{m}+1\right)} x^{j_{m}}\left(1-\sum\limits_{i=1}^{m-1} \omega_{i}^{k_{i}} t_{i}^{1 / r_{i}}\right)^{-\left(1 / p_{m}+u-1\right)}= \\ \left(1-\sum\limits_{i=1}^{m-1} \omega_{i}^{k_{i}} t_{i}^{1 / r_{i}}\right)^{-\left(1 / p_{m}+u-1\right)} \sum\limits_{v=0}^{u-1} b_{v}(u) \sum\limits_{j_{m}=0}^{\infty} \frac{\Gamma\left(j_{m}+v+1\right)}{\Gamma\left(j_{m}+1\right)} x^{j_{m}} 。$

由引理3可得

$\sum\limits_{j_{m}=0}^{\infty} \frac{\Gamma\left(j_{m}+v+1\right)}{\Gamma\left(j_{m}+1\right)} x^{j_{m}}=\frac{\Gamma(v+1)}{(1-x)^{v+1}},$

故有

${\mathit{\Lambda}}\left(t_{1}, \cdots, t_{m}\right)=\frac{p_{1} \cdots p_{m}}{\chi(0) V_{{\mathit{\Omega}}}} \sum\limits_{j=1}^{n+2} b_{j} \sum\limits_{u=0}^{j} b_{u}(j) \times \\ \;\;\;\;\;\; \frac{\partial^{m-1}}{\partial t_{1} \cdots \partial t_{m-1}} \sum\limits_{k_{1}=0}^{r_{1}-1} \cdots \sum\limits_{k_{m-1}=0}^{r_{m-1}{-1}}\left(1-\sum\limits_{i=1}^{m-1} \omega_{i}^{k_{i}} t_{i}^{1 / r_{i}}\right)^{-\left(1 / p_{m}+u-1\right)} \times \\ \;\;\;\;\;\; \sum\limits_{v=0}^{u-1} b_{v}(u) \frac{\Gamma(v+1)}{(1-x)^{v+1}}=\frac{p_1 \cdots p_m }{\chi(0) V_{{\mathit{\Omega}}}} \frac{\partial^{m-1}}{\partial t_1 \cdots \partial t_{m-1}} \times \\ \;\;\;\;\;\; \sum\limits_{k_{1}=0}^{r_{1}-1} \cdots \sum\limits_{k_{m-1}=0}^{r_{m-1}{-1}} \sum\limits_{j=1}^{n+2} \sum\limits_{u=0}^{j} \sum\limits_{v=0}^{u-1} \frac{b_{j} b_{u}(j) b_{v}(u) \Gamma(v+1)}{\left(1-\sum\limits_{i=1}^{m-1} \omega_{i}^{k_{i}} t_{i}^{1 / r_{i}}\right)^{1 / p_{m}+u-1}(1-x)^{v+1}}。$

由于t_i=|W_i^*|²=A(Z, Z)^2/p_i|W_i|²=|W_i|²/N(Z, Z)^1/p_i，则有

$K_{E}\left(W_{1}, \cdots, W_{m}, Z\right)=\\ \;\;\;\;\;\; {\mathit{\Lambda}}\left(\frac{\left|W_{1}\right|^{2}}{N(Z, Z)^{1 / p_{1}}}, \cdots, \frac{\left|W_{m}\right|^{2}}{N(Z, Z)^{1 / p_{m}}}\right) N(Z, Z)^{-\sum\nolimits_{i=1}^{m}{1 / p_{i}-g}}。$

证毕。

由定理1和膨胀原理^[15]可得如下定理：

定理2 域E(q₁, …, q_m, Ω; p₁, …, p_m)的Bergman核函数为：

$K_{E}\left(q_{1}, \cdots, q_{m}, {\mathit{\Omega}} ; p_{1}, \cdots, p_{m}\right)=\frac{1}{q_{1}!\cdots q_{m}!} \times \\ \;\;\;\;\;\;{\mathit{\Lambda}}^{\left(q_{1}-1, \cdots, q_{m}-1\right)}\left(\frac{\left\|W_{1}\right\|^{2}}{N(Z, Z)^{1 / p_{1}}}, \cdots, \frac{\left\|W_{m}\right\|^{2}}{N(Z, Z)^{1 / p_{m}}}\right) \times \\ \;\;\;\;\;\; \quad N(Z, Z)^{-\sum\nolimits_{i=1}^{m} q_{i} / p_{i}-g},$

其中，

${\mathit{\Lambda}}^{\left(q_{1}-1, \cdots, q_{m}-1\right)}\left(t_{1}, \cdots, t_{m}\right)=\frac{\partial^{q_{1}+\cdots+q_{m}-m} {\mathit{\Lambda}}\left(t_{1}, \cdots, t_{m}\right)}{\partial t_{1}^{q_{1}-1} \cdots \partial t_{m}^{q_{m}-1}}$

是Λ的偏导数。

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