$ be a finite point process with density $p$ satisfying $$\label{hered}p({\mathbf x})>0\Rightarrow p(\tilde{{\mathbf x}})>0$$ for all $\tilde{{\mathbf x}}\subset {\mathbf x}.$ For the (Papangelou) conditional intensity of $\mu ,$ see [@RefA], it holds $$\lambda^*(u;{\mathbf x})=\frac{p({\mathbf x}\cup \{u\})}{p({\mathbf x})},\; {\mathbf x}\in {\mathbf N},\: u\in B,\: u\notin {\mathbf x}.$$ For $p({\mathbf x})=0$ we put $\lambda^*(u;{\mathbf x})=0.$ For $n>1$ we use analogously $$\lambda^*_n(u_1,\dots ,u_n;{\mathbf x})=\frac{p({\mathbf x}\cup \{u_1,\dots ,u_n\})}{p({\mathbf x})},$$ $u_1,\dots ,u_n\in B\setminus\{{\mathbf x}\}
$ distinct, the conditional intensity of $n$-th order of $\mu ,\;\lambda^*_0\equiv 1.$ We observe that $\lambda^*_n$ is symmetric in the variables $u_1,\dots ,u_n.$ The expectations of conditional intensities $$\label{coref}\rho_n(u_1,\dots ,u_n;\mu )={\mathbb E}\lambda^*(u_1,\dots ,u_n;\mu )={\mathbb E}[p(\eta\cup\{u_1,\dots ,u_n\})],$$ are called $n-$th correlation functions of the point process $\mu ,$ cf. [@RefG].

For a functional $F,\;y\in B,$ one defines the difference operator $D_yF$ for a point process $\mu $ as a random variable $$D_yF(\mu )=F(\mu + \delta_y)-F(\mu ),$$ where $\delta_y $ is a Dirac measure at the point $y.$ Inductively for $n\geq 2$ and $(y_1,\dots ,y_n)\in B^n$ we define a function $$D_{y_1,\dots ,y_n}^nF=D_{y_1}^1D_{y_2,\dots ,y_n}^{n-1}F,$$ where $D^1_y=D_y,\; D^0F=F.$ Operator $D^n_{y_1,\dots ,y_n}$ is symmetric in $y_1,\dots ,y_n$ and symmetric functions $T_n^\mu F$ on $B^n$ are defined as $$T^\mu_nF(y_1,\dots ,y_n)={\mathbb E}D_{y_1,\dots ,y_n}^nF(\mu ),$$ $n\in {\mathbb N},\; T_0^\mu 