Let $\phi(z)$ be a non-negative function on $\mathbb{C}^d$, and

$\begin{equation}\label{eq1.1} D_{\phi}:=\left\{(z_0,z)\in \mathbb{C}\times\mathbb{C}^d: \textrm{Im} z_0>\phi(z)\right\}. \end{equation}$

(1.1)

The Hardy space $H^2( D_{\phi})$ is defined by

$H^2( D_{\phi}):=\left\{f\in H( D_{\phi}):\sup\limits_{s>0}\int_{\mathbb{R}}\int_{\mathbb{C}^d}|f(z,t+\sqrt{-1}\phi(z)+\sqrt{-1}s)|^2dtdm(z)<+\infty \right\},$

where $H( D_{\phi})$ denotes all holomorphic functions on the domain $ D_{\phi}$, and $dm(z)$ is the Lebesgue measure on $\mathbb{C}^d$. The closed subspace $H^2(\partial D_{\phi})$ of $L^2(\partial D_{\phi})$ consisting of boundary values of holomorphic functions $f\in H^2( D_{\phi})$. The Szegö projection is the orthogonal projection

$S: L^2(\partial D_{\phi})\rightarrow H^2(\partial D_{\phi}),$

and the Szegö kernel $S(z,t;u,s)$ is the distribution kernel on $\partial D_{\phi}\times \partial D_{\phi}$ give by

$\begin{equation}\label{eq1.2} Sf(z,t):=\int_{\partial D_{\phi}}S(z,t;u,s)f(u,s)dm(u)ds, \end{equation}$

(1.2)

where the boundary $\partial D_{\phi}$ of $ D_{\phi}$ be identified with $\mathbb{C}^d\times\mathbb{R}$, the coordinates are $(z,t)$.

Let $d_1, d_2, \cdots, d_n$ be positive integers, $s_1, s_2, \cdots, s_n$ be positive real numbers, $\phi(z)=\sum\limits_{i=1}^n||z_i||^{\frac{2}{s_i}}$ with $z_i=(z_{i1},z_{i2},\cdots,z_{id_i})\in \mathbb{C}^{d_i}$, where $ ||z_i||^2:=\sum\limits_{j=1}^{d_i}|z_{ij}|^2$. In [1], Francsics and Hanges obtained the Szegö kernel of the unbounded domain $ D_{\phi}$. For special $\phi(z)$, the Szegö kernel of $ D_{\phi}$, see also the references of [1].

Let $\Omega_i$ be irreducible bounded symmetric domains (Cartan domains) in $\mathbb{C}^{d_i}$ in its Harish-Chandra realization, $s_i$ be positive real numbers, $1\leq i\leq n$. The following we assume that

$\begin{equation}\label{eq1.3} D_{\psi}:=\left\{(z_0,z)\in \mathbb{C}\times\mathbb{C}^d: \textrm{Im} z_0>\psi(z)\right\}, \end{equation}$

(1.3)

where $\psi(z)=\sum\limits_{i=1}^n\|z_i\|^{\frac{2}{s_i}}$, $z=(z_1,z_2,\cdots,z_n)\in \mathbb{C}^{d_1}\times \mathbb{C}^{d_2}\times \cdots \times \mathbb{C}^{d_n}$, $d=\sum\limits_{j=1}^nd_j$ and $\|z_i\|$ are the spectral norms of $z_i$, see (3.5). If the ranks of $\Omega_i$ equal to 1 for all $1\leq i \leq n$, then here $\psi(z)$ same as the above $\phi(z)$.

For convenience, we list classical domains and corresponding the generic norms $N(z,\overline{z})$ and the spectral norms $\|z\|$ as following (see [2, 3])

$\begin{equation*} \mathfrak{R}_I(m,n)=\{z\in \mathbb{C}^{m\times n}:\|z\|<1\}(m\leq n), N(z,\overline{z})=\det(I-zz^{\dagger}), \|z\|=\sqrt{\sup\limits_{uu^{\dagger}=1}uzz^{\dagger}u^{\dagger}}, \end{equation*}$

where $z^{\dagger}$ denotes the conjugation transposition of $z$. If the rank $m$ of $\mathfrak{R}_I(m,n)$ equal to 1, then $\|z\|=\sqrt{zz^{\dagger}}$.

$\begin{equation*} \mathfrak{R}_{II}(n)=\{z\in \mathbb{C}^{n\times n}:z=z^t,\|z\|<1\}, N(z,\overline{z})=\det(I-zz^{\dagger}),\|z\|=\sqrt{\sup\limits_{uu^{\dagger}=1}uzz^{\dagger}u^{\dagger}}, \end{equation*}$

where $z^t$ denotes the transposition of $z$.

$\begin{eqnarray*} &&\mathfrak{R}_{III}(n)=\{z\in \mathbb{C}^{n\times n}:z=-z^t,\|z\|<1\}, N(z,\overline{z})=\sqrt{\det(I-zz^{\dagger})},\|z\|=\sqrt{\sup\limits_{uu^{\dagger}=1}uzz^{\dagger}u^{\dagger}}, \\ &&\mathfrak{R}_{IV}(n)=\{z\in \mathbb{C}^{n}:\|z\|<1\}, N(z,\overline{z})=1-2zz^{\dagger}+zz^t\overline{zz^t},\|z\|=\sqrt{zz^{\dagger}+\sqrt{(zz^{\dagger})^2-zz^t\overline{zz^t}}}. \end{eqnarray*}$

In this note, by using the method of [1], we will calculate the Szegö kernel of $ D_{\psi}$. In the following Section 2 and Section 3, we collect basic material about the generalized Selberg formula, integrals of Jack polynomials time the certain weight, and integrals of $K_{\lambda}$ over the Cartan domain. In Section 4, we compute the Szegö kernel of $ D_{\psi}$.

Lemma 2.1 (Generalized Selberg formula [4--6])) For give ${\rm Re}(x)>-1, {\rm Re}(y)>-1, \alpha>0$, let $\lambda$ be any partition of length $\ell(\lambda)\leq n$, $P_{\lambda}^{(\alpha)}$ be the symmetric Jack polynomial, then we have

$\begin{eqnarray} \nonumber & & \int_{[0,1]^n}P_{\lambda}^{(\alpha)}(x_1,x_2,\cdots,x_n)\prod\limits_{i=1}^nx_i^{x}(1-x_i)^{y}\prod\limits_{1\leq j<k \leq n}|x_j-x_k|^{\frac{2}{\alpha}}\prod\limits_{j=1}^ndx_j \\ \label{1-10} &=&{P}_{\lambda}^{(\alpha)}(1_n)\frac{[x+1+\frac{n-1}{\alpha}]_{\lambda}^{(\alpha)}}{[x+y+2+2\frac{n-1}{\alpha}]_{\lambda}^{(\alpha)}}S_n(x,y;\alpha), \end{eqnarray}$

(2.1)

where

$\begin{equation}\label{1-11} S_n(x,y;\alpha)=\prod\limits_{j=0}^{n-1}\frac{\Gamma(x+1+\frac{j}{\alpha})\Gamma(y+1+\frac{j}{\alpha})\Gamma(1+\frac{j+1}{\alpha})}{\Gamma(x+y+2+\frac{n+j-1}{\alpha})\Gamma(1+\frac{1}{\alpha})} \end{equation}$

(2.2)

and

$[s]^{(\alpha)}_{\lambda}:=\prod\limits_{j=1}^{\ell(\lambda)}\left(s-\frac{j-1}{\alpha}\right )_{\lambda_j}, $

${P}_{\lambda}^{(\alpha)}(1_n)$ denotes the value of the function ${P}_{\lambda}^{(\alpha)}(x_1,x_2,\cdots,x_n)$ at $(x_1,x_2,\cdots,x_n)=(1,1,\cdots,1)$.

The following we compute the integral of Jack polynomial times the certain weight using the generalized Selberg formula, to this end, we first give the following results.

Lemma 2.2 Let $\varphi(x_1,x_2,\cdots,x_n)$ and $f(x_1,x_2,\cdots,x_n)$ be continuous real functions such that

(1) $\forall x_i\geq 0, 1\leq i \leq n$, $\varphi(x)\geq 0$. $\varphi(x)=0$ if and only if $x=0$.

(2) $\forall \alpha\in \mathbb{R}, \varphi(\alpha x)=|\alpha|\varphi(x)$.

(3) $\forall t\in \mathbb{R}$, $f(tx)=t^df(x)$. $\forall x_i\geq 0, 1\leq i \leq n$, $f(x)\geq 0$. Then for all positive real numbers $s, s_1, s_2$, we have

(1)

$\begin{equation}\label{e1} \int_{x\geq 0 \atop \varphi(x)<1}f(x)\left (1-\varphi(x)^{\frac{1}{s_2}}\right )^{s_1}dx=\frac{\Gamma(s_1+1)\Gamma(s_2(d+n)+1)}{\Gamma(s_1+s_2(d+n)+1)}\int_{x\geq 0 \atop \varphi(x)<1}f(x)dx, \end{equation}$

(2.3)

where $x=(x_1, x_2, \cdots, x_n)$, $dx:=dx_1dx_2\cdots dx_n$, and $x\geq 0$ mean $\forall i, 1\leq i \leq n, x_i\geq 0$.

(2)

$\begin{equation}\label{e2} \int_{x\geq 0}f(x)\exp\{-\varphi(x)^{\frac{1}{s}}\}dx=\Gamma(s(d+n)+1)\int_{x\geq 0 \atop \varphi(x)<1}f(x)dx. \end{equation}$

(2.4)

Proof (1) Let

$\begin{equation}\label{e4} I=\int_{u+\varphi(x)^{\frac{1}{s_2}}<1\atop u\geq 0, x\geq 0}u^{s_1-1}f(x)dudx. \end{equation}$

(2.5)

On the one hand,

$\begin{eqnarray} \nonumber I &=& \int_0^1 u^{s_1-1}du\int_{\varphi(x)<(1-u)^{s_2}\atop x\geq 0}f(x)dx= \int_0^1 u^{s_1-1} (1-u)^{s_2(d+n)}du\int_{\varphi(x)<1\atop x\geq 0}f(x)dx \\ \label{e5} &=& \frac{\Gamma(s_1)\Gamma(s_2(d+n)+1)}{\Gamma(s_1+s_2(d+n)+1)}\int_{\varphi(x)<1\atop x\geq 0}f(x)dx . \end{eqnarray}$

(2.6)

On the other hand,

$\begin{eqnarray} \nonumber I &=& \int_{\varphi(x)<1\atop x\geq 0}f(x)dx \int_{0\leq u<1-\varphi(x)^{\frac{1}{s_2}}} u^{s_1-1}du=\int_{\varphi(x)<1\atop x\geq 0}f(x)(1-\varphi(x)^{\frac{1}{s_2}})^{s_1}dx \int_0^1 u^{s_1-1} du\\ \label{e6} &=& \frac{1}{s_1} \int_{\varphi(x)<1\atop x\geq 0}f(x)(1-\varphi(x)^{\frac{1}{s_2}})^{s_1}dx. \end{eqnarray}$

(2.7)

By (2.6) and (2.7), we obtain (2.3).

(2) In (2.3), we set $s_1=m^{\frac{1}{s}}, s_2=s$, by the change of variables $x=\frac{y}{m}, m>0$, we have

$\begin{equation}\label{e7} \int_{y\geq 0 \atop \varphi(y)<m}f(y)\left (1-\frac{1}{m^{\frac{1}{s}}}\varphi(y)^{\frac{1}{s}}\right )^{m^{\frac{1}{s}}}dy=m^{d+n}\frac{\Gamma(m^{\frac{1}{s}}+1)\Gamma(s(d+n)+1)}{\Gamma(m^{\frac{1}{s}}+s(d+n)+1)}\int_{x\geq 0 \atop \varphi(x)<1}f(x)dx. \end{equation}$

(2.8)

When $m\rightarrow +\infty$, limit of L.H.S. of (2.8) is $\displaystyle \int_{y\geq 0}f(y)\exp\{-\varphi(y)^{\frac{1}{s}}\}dy,$ and limit of R.H.S. of (2.8) is

$\Gamma(s(d+n)+1)\int_{x\geq 0 \atop \varphi(x)<1}f(x)dx.$

Here we use $\frac{\Gamma(x+a)}{\Gamma(x+b)}\sim x^{a-b}(x\rightarrow +\infty).$ So we obtain (2.4). The proof is completed.

For $s>0, x=(x_1,x_2,\cdots,x_n)$, setting $\|x\|_s:=(\sum\limits_{i=1}^n|x_i|^s)^{\frac{1}{s}}$. It is easy to see $\|x\|_1:=\sum\limits_{i=1}^n|x_i|$ and $\|x\|_{\infty}=\max\limits_{1\leq i\leq n}\{|x_i|\}$. Let

$P(\lambda,\alpha,b,n,x):=P_{\lambda}^{(\alpha)}(x_1,x_2,\cdots,x_n)\prod\limits_{i=1}^nx_i^{b}\prod\limits_{1\leq j<k \leq n}|x_j-x_k|^{\frac{2}{\alpha}}.$

Since homogeneous the function $P(\lambda,\alpha,b,n,x)$ of degree $d=|\lambda|+nb+\frac{n(n-1)}{\alpha}$ satisfies Lemma 2.2. By Lemma 2.2, we have the following Corollary 2.3.

Corollary 2.3 For all positive real numbers $s, t$, we have

$\begin{eqnarray} \nonumber & & \int_{x\geq 0}P(\lambda,\alpha,b,n,x)\exp\{-\|x\|_s^{\frac{1}{t}}\}dx \\ \label{e11} &=& \Gamma(t(|\lambda|+nb+\frac{n(n-1)}{\alpha}+n)+1)\int_{x\geq 0 \atop \|x\|_s<1}P(\lambda,\alpha,b,n,x)dx. \end{eqnarray}$

(2.9)

As a consequence of Lemma 2.1, we have

$\begin{eqnarray} \nonumber & & \int_{x\geq 0\atop \|x\|_{\infty}\leq 1}P(\lambda,\alpha,b,n,x)\\ \label{e13} &=& P_{\lambda}^{(\alpha)}(1_n)\frac{[b+1+\frac{n-1}{\alpha}]_{\lambda}^{(\alpha)}}{[b+2+2\frac{n-1}{\alpha}]_{\lambda}^{(\alpha)}} \prod\limits_{j=0}^{n-1}\frac{\Gamma(b+1+\frac{j}{\alpha})\Gamma(1+\frac{j}{\alpha})\Gamma(1+\frac{j+1}{\alpha})}{\Gamma(b+2+\frac{n+j-1}{\alpha})\Gamma(1+\frac{1}{\alpha})}. \end{eqnarray}$

(2.10)

From Corollary 2.3 and (2.10) we get

Corollary 2.4 For all positive real numbers $s$, we have

$\begin{eqnarray} \nonumber & & \int_{x\geq 0}P(\lambda,\alpha,b,n,x)\exp\{-\|x\|_{\infty}^{\frac{1}{s}}\}dx \\ \label{e15} &=& \Gamma(s(|\lambda|+nb+\frac{n(n-1)}{\alpha}+n)+1)P_{\lambda}^{(\alpha)}(1_n)\frac{[b+1+\frac{n-1}{\alpha}]_{\lambda}^{(\alpha)}}{[b+2+2\frac{n-1}{\alpha}]_{\lambda}^{(\alpha)}}S_n(b,0;\alpha). \end{eqnarray}$

(2.11)

Let $\Omega\subset \mathbb{C}^{d}$ be Cartan domain, we denote by $r, a, b, d, p $ and $N(z,\overline{w})$ the rank, the characteristic multiplicities, the dimension, the genus, and the generic norm of $\Omega$, respectively. Let $\mathcal{G}$ stand for the identity connected component of the group of biholomorphic self-maps of $\Omega$, and $\mathcal{K}$ {for the stabilizer} of the origin in $\mathcal{G}$.

Under the action $f\mapsto f\circ k (k\in \mathcal{K})$ of $\mathcal{K}$, the space $\mathcal{P}$ of holomorphic polynomials on $\mathbb{C}^d$ admits the Peter-Weyl decomposition

$\mathcal{P}=\bigoplus_{\lambda}\mathcal{P}_{\lambda},$

where the spaces $\mathcal{P}_{\lambda}$ are $\mathcal{K}$-invariant and irreducible.

Let

$\begin{equation}\label{1.4} {\langle}f,g {\rangle}_{\mathcal{F}}:=\frac{1}{\pi^d}\int_{\mathbb{C}^d}f(z)\overline{g(z)}e^{-|z|^2}dm(z), \end{equation}$

(3.1)

where $dm(z)$ denotes the Lebesgue measure on $\mathbb{C}^d$.

For every partition $\lambda$, let $K_{\lambda}(z_1,\overline{z_2})$ be the reproducing kernel of $\mathcal{P}_{\lambda}$ with respect to (3.1). The kernels $K_{\lambda}(z_1,\overline{z_2})$ are related to the generic norm $N(z_1,\overline{z_2})$ by the Faraut-{Korányi} formula

$\begin{equation}\label{1.7} N(z_1,\overline{z_2})^{-s}=\sum\limits_{\lambda}(s)_{\lambda}K_{\lambda}(z_1,\overline{z_2}), \end{equation}$

(3.2)

where $(s)_{\lambda}$ denotes the generalized Pochhammer symbol

$\begin{equation}\label{1.8} (s)_{\lambda}:=\prod\limits_{j=1}^r\left(s-\frac{j-1}{2}a\right )_{\lambda_j}. \end{equation}$

(3.3)

Here $(s)_m$ denotes the raising factorial

$ {(s)_m:=\frac{\Gamma(s+m)}{\Gamma(s)}=s(s+1)\cdots (s+m-1)}.$

Let $e_1,e_2,\cdots, e_r\in\mathbb{C}^d$ be a Jordan frame. Then each $z\in \mathbb{C}^d$ has the polar decomposition

$\begin{equation}\label{1.9} z=k\cdot(t_1e_1+t_2e_2+\cdots+t_re_r),\quad k\in \mathcal{K}, t_1\geq t_2\geq \cdots \geq t_r\geq 0. \end{equation}$

(3.4)

The numbers $t_1,t_2,\cdots, t_r$ are called the singular values of $z$. the spectral norm of $z$ is defined by

$\begin{equation}\label{1.10} \|z\|:=\max\{t_1,t_2,\cdots, t_r\}. \end{equation}$

(3.5)

It is known that

$\begin{equation}\label{1.11} K_{\lambda}(z,\overline{z})= K_{\lambda}(\sum\limits_{j=1}^rt_j^2e_j,\overline{e}), \end{equation}$

(3.6)

For the proofs of above facts and additional details, we refer e.g. to [7].

Lemma 3.1 For give a positive real number $s$, we have

$\begin{equation}\label{e18} \int_{\mathbb{C}^d}K_{\lambda}(z,\overline{z})\exp\{-\|z\|^{\frac{2}{s}}\}dm(z)=\Gamma(s(|\lambda|+d)+1) \frac{\dim \mathcal{P}_{\lambda}}{(p)_{\lambda}}V(\Omega), \end{equation}$

(3.7)

where $V(\Omega)$ is the volume with respect to the Euclidean measure of $\Omega$.

Proof We recall the formula for integration in polar coordinates (see [7])

$\begin{equation}\label{e20} \int_{\mathbb{C}^d}f(z)dm(z)=c\int_{[0,+\infty)^r}2^r\prod\limits_{j=1}^rt_j^{2b+1}\prod\limits_{1\leq j<k \leq r}|t_j^2-t_k^2|^a\prod\limits_{j=1}^rdt_j\int_{\mathcal{K}}f(k\cdot\sum\limits_{j=1}^rt_je_j)dk, \end{equation}$

(3.8)

where $c$ is a constant. By using (3.8), (3.5) and (3.6), we have

$\begin{equation}\label{e22} \text{L.H.S. of (3.7)}=c\int_{[0,+\infty)^r}K_{\lambda}(\sum\limits_{j=1}^rt_je_j,\overline{e})\exp\{-\|t\|_{\infty}^{\frac{1}{s}}\}\prod\limits_{j=1}^rt_j^b\prod\limits_{1\leq j<k\leq r}|t_j-t_k|^a\prod\limits_{j=1}^rdt_j. \end{equation}$

(3.9)

For each partition $\lambda$, since the polynomial $K_{\lambda}(\sum\limits_{j=1}^rt_je_j,\overline{e})$ in $t_1,t_2,\cdots,t_r$ is proportional to the Jack polynomial $P_{\lambda}^{(\frac{2}{a})}(t_1,t_2,\cdots,t_r)$, by (2.11) for (3.9), we get

$\begin{equation}\label{e25} \text{L.H.S. of (3.7)}=\Gamma(s(|\lambda|+d)+1) cS_r(b,0;{2}/{a})\frac{K_\lambda(e,\overline{e})(\frac{d}{r})_\lambda}{(p)_\lambda}, \end{equation}$

(3.10)

where we have used $d=rb+\frac{a}{2}r(r-1)+r, p=(r-1)a+b+2, (s)_{\lambda}=[s]^{(\frac{2}{a})}_{\lambda}$.

It is well known that (see [8])

$\begin{equation}\label{e27} \dim \mathcal{P}_{\lambda}=K_\lambda(e,\overline{e})(\frac{d}{r})_\lambda. \end{equation}$

(3.11)

Applying (3.8) we obtain

$\begin{equation}\label{e28} V(\Omega):= \int_{\Omega}dm(z)=cS_r(b,0;{2}/{a}). \end{equation}$

(3.12)

Substituting (3.11), (3.12) into (3.10), we have (3.7). The proof is completed.

Let $\|z_i\|$ be the spectral norms of Carta domains $\Omega_i(1\leq i \leq n)$, in this section, we will calculate the Szegö kernel of a domain $ D_{\psi}$

$\begin{equation*} D_{\psi}:=\left\{(z_0,z)\in \mathbb{C}\times\mathbb{C}^d: \textrm{Im} z_0>\psi(z)\right\}, \end{equation*}$

where

$\begin{equation*} \psi(z):=\sum\limits_{i=1}^n\|z_i\|^{\frac{2}{s_i}},z=(z_1,z_2,\cdots,z_n),d=\sum\limits_{j=1}^n\dim\Omega_j. \end{equation*}$

Theorem 4.1 Let $r_i, a_i, b_i, d_i, p_i, (s)_\lambda^{(i)}$, $V(\Omega_i)$ and $N_i$ be ranks, characteristic multiplicities, dimensions, genuses, generalized Pochhammer symbols, volumes and generic norms of Cartan domains $\Omega_i$, $1\leq i \leq n$, respectively. The Szegö kernel of $ D_{\psi}$ is

$\begin{eqnarray} \nonumber & & S(z_0,t_1z_1,\cdots,t_nz_n;\overline{u_0},\overline{u_1},\cdots,\overline{u_n}) \\ \label{e4.1} &=& \frac{1}{4\pi}\prod\limits_{i=1}^n\frac{1}{V(\Omega_i)} \frac{1}{A^{1+\sum\limits_{i=1}^ns_id_i}}\varphi(t_1\frac{d}{dt_1},\cdots,t_n\frac{d}{dt_n}) \prod\limits_{i=1}^n \left\{ \frac{1}{N_i(\frac{t_iz_i}{A^{s_i}},\overline{u_i})}\right\}^{p_i}, \end{eqnarray}$

(4.1)

where $\forall 1\leq i\leq n, t_i\in [0,1]$, the function $\varphi(t_1,\cdots,t_n)$ is given

$\begin{equation}\label{e4.2} \varphi(t_1,\cdots,t_n)=\frac{\Gamma(\sum\limits_{i=1}^ns_i(d_i+t_i)+1)}{\prod\limits_{i=1}^n\Gamma(s_i(d_i+t_i)+1)}, \end{equation}$

(4.2)

$\varphi(t_1\frac{d}{dt_1},\cdots,t_n\frac{d}{dt_n})$ is defined by

$\begin{equation}\label{e4.2.1} \varphi(t_1\frac{d}{dt_1},\cdots,t_n\frac{d}{dt_n})t_1^{i_1}\cdots t_n^{i_n}:=\varphi(i_1,\cdots,i_n)t_1^{i_1}\cdots t_n^{i_n}, \end{equation}$

(4.3)

here $i_1,\cdots,i_n\in \mathbb{N}$, and

$\begin{equation}\label{e4.3} A=-\frac{\sqrt{-1}}{2}(z_0-\overline{u_0})=\frac{1}{2}\left(\sum\limits_{i=1}^n\left(\|z_i\|^{\frac{2}{s_i}}+\|u_i\|^{\frac{2}{s_i}}\right)-\sqrt{-1}\textrm{Re}(z_0-u_0)\right). \end{equation}$

(4.4)

In particular for $n=1$, the Szegö kernel of $ D_{\psi}$ is

$\begin{equation}\label{e4.1.1} S(z_0,z_1;\overline{u_0},\overline{u_1})= \frac{1}{4\pi V(\Omega_1)A^{1+s_1d_1}} \left\{ \frac{1}{N_1(\frac{z_1}{A^{s_1}},\overline{u_1})}\right\}^{p_1}. \end{equation}$

(4.5)

Proof By [1], the Szegö kernel of $ D_{\psi}$ is written as

$\begin{equation}\label{e4.4} S(z_0,z;\overline{u_o},\overline{u})=\int_0^{+\infty}\exp\{-4\pi tA\} K_t(z,\overline{u})dt, \end{equation}$

(4.6)

where $K_t(z,\overline{u})$ is the reproducing kernels of the Hilbert spaces $L_a^2(\mathbb{C}^d,\rho_t)$, here

$L_a^2(\mathbb{C}^d,\rho_t):=\{f\in H(\mathbb{C}^d)|(f,f)<+\infty\},\quad \rho_t(z):=\exp\{-4\pi t\phi(z)\}, $

where $H(\mathbb{C}^d)$ denotes the space of holomorphic functions on $\mathbb{C}^d$, and the inner product $(\cdot,\cdot)$ is defined by

$(f,g):=\int_{\mathbb{C}^d}f(z)\overline{g(z)}\rho_t(z)dm(z).$

Let $\mathcal{G}_i$ stand for the identity connected components of groups of biholomorphic self-maps of $\Omega_i \subset \mathbb{C}^{d_i}$, and $\mathcal{K}_i$ for stabilizers of the origin in $\mathcal{G}_i$, $1\leq i \leq n$, respectively. For any $k=(k_1,\cdots, k_n)\in \mathcal{K}:= \mathcal{K}_1\times\cdots\times\mathcal{K}_n$, we define the action

$ \pi(k)f(z_1,\cdots,z_n)\equiv f\circ k(z_1,\cdots,z_n):=f(k_1\circ z_1,\cdots,k_n\circ z_n)$

of $\mathcal{K}$, then the space $\mathcal{P}$ of holomorphic polynomials on $\mathbb{C}^{d_1}\times\cdots\times\mathbb{C}^{d_n}$ admits the decomposition

$\mathcal{P}=\bigoplus_{{\ell(\lambda^i)\leq r_i\atop 1\leq i\leq n}}\mathcal{P}^{(1)}_{\lambda^1}\otimes\cdots\otimes \mathcal{P}^{(n)}_{\lambda^n},$

where spaces $\mathcal{P}^{(i)}_{\lambda^i}$ are $\mathcal{K}_i$-invariant and irreducible subspaces of spaces of holomorphic polynomials on $\mathbb{C}^{d_i}(1\leq i \leq n)$.

Since $\mathbb{C}^d=\mathbb{C}^{d_1}\times\cdots\times\mathbb{C}^{d_n}$ invariant under the action of $\mathcal{K}_1\times\cdots\times\mathcal{K}_n$, $L_a^2(\mathbb{C}^d,\rho_t)$ admits an irreducible decomposition (see ref. [9])

$L_a^2(\mathbb{C}^d,\rho_t)=\widehat{\bigoplus_{{\ell(\lambda^i)\leq r_i\atop 1\leq i\leq n}}}\mathcal{P}^{(1)}_{\lambda^1}\otimes\cdots\otimes \mathcal{P}^{(n)}_{\lambda^n},$

where $\widehat{\bigoplus}$ denotes the orthogonal direct sum.

For every partition $\lambda^i$ of length less than or equal to $r_i$, let $K^{(i)}_{\lambda^i}(z_i,\overline{u_i})$ be the reproducing kernels of $\mathcal{P}^{(i)}_{\lambda^i}$ with respect to (3.1). By Schur's lemma, there exist positive constants $c_{\lambda^1\cdots\lambda^n}$ such that $c_{\lambda^1\cdots\lambda^n}\prod\limits_{i=1}^nK^{(i)}_{\lambda^i}(z_i,\overline{u_i})$ is reproducing kernels of $\mathcal{P}^{(1)}_{\lambda^1}\otimes\cdots\otimes \mathcal{P}^{(n)}_{\lambda^n}$ with respect to the above inner product $(\cdot,\cdot)$. According to the definition of reproducing kernel, we have

$\int_{\mathbb{C}^d}c_{\lambda^1\cdots\lambda^n}\prod\limits_{i=1}^nK^{(i)}_{\lambda^i}(z_i,\overline{z_i})\rho_t(z_1,\cdots,z_n)\prod\limits_{i=1}^ndm(z_i)=\prod\limits_{i=1}^n\dim\mathcal{P}^{(i)}_{\lambda^i}.$

Therefore, the reproducing kernels of $L_a^2(\mathbb{C}^d,\rho_t)$ can be written as

$\begin{equation}\label{e4.5} K_t(z_1,\cdots,z_n;\overline{u_1},\cdots,\overline{u_n})=\sum\limits_{{\ell(\lambda^i)\leq r_i\atop 1\leq i\leq n}}\frac{\prod\limits_{i=1}^n\dim\mathcal{P}^{(i)}_{\lambda^i}}{<\prod\limits_{i=1}^nK^{(i)}_{\lambda^i}(z_i,\overline{z_i})>}\prod\limits_{i=1}^nK^{(i)}_{\lambda}(z_i,\overline{u_i}), \end{equation}$

(4.7)

where $<f>$ denotes integral

$\int_{\mathbb{C}^d}f(z_1,\cdots,z_n)\rho_t(z_1,\cdots,z_n)\prod\limits_{i=1}^ndm(z_i).$

From (3.7), we have

$\begin{eqnarray} \nonumber & & \int_{\mathbb{C}^d}\prod\limits_{i=1}^nK^{(i)}_{\lambda^i}(z_i,\overline{z_i})\rho_t(z_1,\cdots,z_n)\prod\limits_{i=1}^ndm(z_i) \\ \label{e4.6} &=& \prod\limits_{i=1}^n(4\pi t)^{-s_i(|\lambda^i|+d_i)}\Gamma(s_i(|\lambda^i|+d_i)+1)\frac{\dim \mathcal{P}^{(i)}_{\lambda^i}}{(p_i)^{(i)}_{\lambda^i}}V(\Omega_i). \end{eqnarray}$

(4.8)

It follows from (4.7) and (4.8) that

$\begin{eqnarray} \nonumber & & K_t(z_1,\cdots,z_n;\overline{u_1},\cdots,\overline{u_n}) \\ \label{e4.7} &=& \sum\limits_{{\ell(\lambda^i)\leq r_i\atop 1\leq i\leq n}}(4\pi t)^{\sum\limits_{i=1}^ns_i(|\lambda^i|+d_i)} \prod\limits_{i=1}^n\frac{1}{\Gamma(s_i(|\lambda^i|+d_i)+1)V(\Omega_i)} \prod\limits_{i=1}^n(p_i)^{(i)}_{\lambda^i}K^{(i)}_{\lambda}(z_i,\overline{u_i}). \end{eqnarray}$

(4.9)

Now substituting (4.9) into (4.6), by

$\begin{equation}\label{e4.8} \int_0^{+\infty}\exp\{-4\pi tA\}(4\pi t)^{\sum\limits_{i=1}^ns_i(|\lambda^i|+d_i)}dt=\frac{1}{4\pi}A^{-\sum\limits_{i=1}^ns_i(|\lambda^i|+d_i)-1}\Gamma(\sum\limits_{i=1}^ns_i(|\lambda^i|+d_i)+1), \end{equation}$

(4.10)

and (3.2), we get

$\begin{eqnarray} \nonumber & & S(z_0,t_1z_1,\cdots,t_n z_n;\overline{u_o},\overline{u_1},\cdots,\overline{u_n})\\ \nonumber &=& \frac{1}{4\pi}\prod\limits_{i=1}^n\frac{1}{V(\Omega_i)} \frac{1}{A^{1+\sum\limits_{i=1}^ns_id_i}} \sum\limits_{{\ell(\lambda^i)\leq r_i\atop 1\leq i\leq n}}\varphi(|\lambda^1|,\cdots,|\lambda^n|) \prod\limits_{i=1}^n(p_i)^{(i)}_{\lambda^i}t_i^{|\lambda^i|}K^{(i)}_{\lambda}( \frac{z_i}{A^{s_i}},\overline{u_i})\\ \nonumber &=& \frac{1}{4\pi}\prod\limits_{i=1}^n\frac{1}{V(\Omega_i)} \frac{1}{A^{1+\sum\limits_{i=1}^ns_id_i}} \sum\limits_{{\ell(\lambda^i)\leq r_i\atop 1\leq i\leq n}}\varphi(t_1\frac{d}{dt_1},\cdots,t_n\frac{d}{dt_n}) \prod\limits_{i=1}^n(p_i)^{(i)}_{\lambda^i}t_i^{|\lambda^i|}K^{(i)}_{\lambda}( \frac{z_i}{A^{s_i}},\overline{u_i})\\ \nonumber &=& \frac{1}{4\pi}\prod\limits_{i=1}^n\frac{1}{V(\Omega_i)} \frac{1}{A^{1+\sum\limits_{i=1}^ns_id_i}} \varphi(t_1\frac{d}{dt_1},\cdots,t_n\frac{d}{dt_n}) \prod\limits_{i=1}^n \left\{ \frac{1}{N_i(\frac{t_iz_i}{A^{s_i}},\overline{u_i})}\right\}^{p_i}, \end{eqnarray}$

where $ 0\leq t_i\leq 1$, this proves Theorem 4.1.

To provide a concrete expression of (4.1), we need Lemma 4.2 below.

Lemma 4.2 If $n>1$, $s_i \in \mathbb{N}_{+}$ and $N_i(z_i,\overline{u_i})=1-z_iu_i^{\dagger}$ for $1\leq i \leq n-1$, $\varphi$ same as (4.2), let $\partial_x=\frac{1}{\prod\limits_{i=1}^{n-1}s_id_i!}\frac{\partial^{d_1+\cdots+d_{n-1}}}{\partial x_1^{d_1}\cdots\partial x_n^{d_{n-1}}}$, then we have

$\begin{eqnarray} \nonumber & & \left.\varphi(t_1\frac{d}{dt_1},\cdots,t_n\frac{d}{dt_n})\left\{\prod\limits_{i=1}^n\frac{1}{N_i(t_iz_i,\overline{u_i})^{p_i}}\right\}\right|_{t_1=\cdots=t_n=1}\\ \nonumber &=&tial_x\sum\limits_{j_1=0}^{s_1-1}\cdots\sum\limits_{j_{n-1}=0}^{s_{n-1}-1}\frac{1}{\left(1-\sum\limits_{i=1}^{n-1}\omega_i^{j_i}x_i^{{1}/{s_i}}\right)^{s_nd_n+1}}\left\{\frac{1}{N_n\left(\frac{z_n}{\left(1-\sum\limits_{i=1}^{n-1}\omega_i^{j_i}x_i^{{1}/{s_i}}\right)^{s_n}},\overline{u_n}\right)}\right\}^{p_n}, \end{eqnarray}$

where $x_i=z_iu_i^{\dagger}$, $\omega_i=\exp\{\frac{2\pi\sqrt{-1}}{s_i}\}$ and symbols $u_i^{\dagger}$ denote the conjugation transposition of the row vectors $u_i$.

Proof For Cartan domains $\Omega_i (1\leq i \leq n-1)$, its ranks $r_i=1$, genuses $p_i=d_1+1$, and the reproducing kernels $K_{k}^{(i)}=\frac{(z_iu_i^{\dagger})^k}{k!}$ of $\mathcal{P}^{(i)}_{k}$ with respect to (3.1), we have

$\begin{eqnarray} \nonumber & & \left.\varphi(t_1\frac{d}{dt_1},\cdots,t_n\frac{d}{dt_n})\left\{\prod\limits_{i=1}^n\frac{1}{N_i(t_iz_i,\overline{u_i})^{p_i}}\right\}\right|_{t_1=\cdots=t_n=1} \\ \nonumber &=& \sum\limits_{\ell(\lambda)\leq r_n}\sum\limits_{k_i=0 \atop 1\leq i \leq n-1}^{+\infty} \varphi(k_1,\cdots,k_{n-1},|\lambda|)\prod\limits_{i=1}^{n-1}\frac{(k_i+1)_{d_i}(z_iu_i^{\dagger})^{k_i}}{d_i!}(p_n)_{\lambda}^{(n)}K^{(n)}_{\lambda}(z_n,\overline{u_n})\\ \nonumber &=&{\prod\limits_{i=1}^{n-1}s_i}\sum\limits_{\ell(\lambda)\leq r_n}\partial_x \sum\limits_{k_i=0 \atop 1\leq i \leq n-1 }^{+\infty} \varphi(k_1,\cdots,k_{n-1},|\lambda|)\prod\limits_{i=1}^{n-1}x_i^{k_i+d_i}(p_n)_{\lambda}^{(n)}K^{(n)}_{\lambda}(z_n,\overline{u_n})\\ \nonumber &=&{\prod\limits_{i=1}^{n-1}s_i}\partial_x\sum\limits_{\ell(\lambda)\leq r_n}\sum\limits_{k_i=0 \atop 1\leq i \leq n-1 }^{+\infty} \frac{\Gamma(\sum\limits_{i=1}^{n-1}s_ik_i+s_n(d_n+|\lambda|)+1)}{\prod\limits_{i=1}^{n-1}\Gamma(s_ik_i+1)\Gamma(s_n(d_n+|\lambda|)+1)}\prod\limits_{i=1}^{n-1}x_i^{k_i} (p_n)_{\lambda}^{(n)}K^{(n)}_{\lambda}(z_n,\overline{u_n}), \end{eqnarray}$

where $x_i=z_iu_i^{\dagger}$.

Let

$\omega_i:=\exp\{\frac{2\pi\sqrt{-1}}{s_i}\}(1\leq i\leq n-1),$

using (3.2), we obtain

$\begin{eqnarray} \nonumber & & \left.\varphi(t_1\frac{d}{dt_1},\cdots,t_n\frac{d}{dt_n})\{\prod\limits_{i=1}^n\frac{1}{N_i(t_iz_i,\overline{u_i})^{p_i}}\}\right|_{t_1=\cdots=t_n=1} \\ \nonumber &=&\partial_x\sum\limits_{j_1=0}^{s_1-1}\cdots\sum\limits_{j_{n-1}=0}^{s_{n-1}-1} \sum\limits_{\ell(\lambda)\leq r_n}\sum\limits_{k_i=0\atop 1\leq i \leq n-1}^{+\infty} \frac{\Gamma(\sum\limits_{i=1}^{n-1}k_i+s_n(d_n+|\lambda|)+1)}{\prod\limits_{i=1}^{n-1}\Gamma(k_i+1)\Gamma(s_n(d_n+|\lambda|)+1)}\\ \nonumber & &\times \prod\limits_{i=1}^{n-1}(\omega_i^{j_i}x_i^{\frac{1}{s_i}})^{k_i} (p_n)_{\lambda}^{(n)}K^{(n)}_{\lambda}(z_n,\overline{u_n})\\ \nonumber &=&\partial_x\sum\limits_{j_1=0}^{s_1-1}\cdots\sum\limits_{j_{n-1}=0}^{s_{n-1}-1} \sum\limits_{\ell(\lambda)\leq r_n}\frac{1} {(1-\sum\limits_{i=1}^{n-1}\omega_i^{j_i}x_i^{{1}/{s_i}})^{s_n(d_n+|\lambda|)+1}}(p_n)_{\lambda}^{(n)}K^{(n)}_{\lambda}(z_n,\overline{u_n})\\ \nonumber &=&\partial_x\sum\limits_{j_1=0}^{s_1-1}\cdots\sum\limits_{j_{n-1}=0}^{s_{n-1}-1}\frac{1}{(1-\sum\limits_{i=1}^{n-1} \omega_i^{j_i}x_i^{{1}/{s_i}})^{s_nd_n+1}}\{\frac{1}{N_n(\frac{z_n}{(1 -\sum\limits_{i=1}^{n-1}\omega_i^{j_i}x_i^{{1}/{s_i}})^{s_n}},\overline{u_n})}\}^{p_n}. \end{eqnarray}$

It completes the proof of Lemma 4.2.

By Lemma 4.2, we give the Szegö kernel of $ D_{\psi}$ in explicit form.

Corollary 4.3 For $s_i, n \in \mathbb{N}_{+}$, $n>1$, and ranks of Cartan domains $\Omega_i$ equal to $1$ $(1\leq i \leq n-1)$, the Szegö kernel of $ D_{\psi}$ may be written as

$\begin{eqnarray} \nonumber & & S(z_0,z_1,\cdots,z_n;\overline{u_0},\overline{u_1},\cdots,\overline{u_n}) \\ \nonumber &=& \frac{1}{4\pi}\prod\limits_{i=1}^n\frac{1}{V(\Omega_i)} \frac{1}{A^{1+\sum\limits_{i=1}^ns_id_i}}\prod\limits_{i=1}^{n-1} \frac{1}{s_id_i!} \frac{\partial^{d_1+\cdots+d_{n-1}}}{\partial x_1^{d_1}\cdots\partial x_n^{d_{n-1}}} \\ \nonumber & & \sum\limits_{j_1=0}^{s_1-1}\cdots\sum\limits_{j_{n-1}=0}^{s_{n-1}-1}\frac{1}{(1-\sum\limits_{i=1}^{n-1}\omega_i^{j_i}x_i^{{1}/{s_i}})^{s_nd_n+1}}\{\frac{1}{N_n(\frac{z_n}{A^{s_n}(1-\sum\limits_{i=1}^{n-1}\omega_i^{j_i}x_i^{{1}/{s_i}})^{s_n}},\overline{u_n})}\}^{p_n}, \end{eqnarray}$

where $ A=-\frac{\sqrt{-1}}{2}(z_0-\overline{u_0})$, $x_i=\frac{z_iu_i^{\dagger}}{A^{s_i}}$, $\omega_i=\exp\{\frac{2\pi\sqrt{-1}}{s_i}\}$ and symbols $u_i^{\dagger}$ denote the conjugation transposition of the row vectors $u_i$.