We define and study a spatial (infinite-dimensional) counterpart of Stirling numbers. In classical combinatorics, the Pochhammer symbol $(m)_n$ can be extended from a natural number $m\in\mathbb N$ to the falling factorials $(z)_n=z(z-1)\dotsm (z-n+1)$ of an argument $z$ from $\mathbb F=\mathbb R\text{ or }\mathbb C$, and Stirling numbers of the first and second kinds are the coefficients of the expansions of $(z)_n$ through $z^k$, $k\leq n$ and vice versa. When taking into account spatial positions of elements in a locally compact Polish space $X$, we replace $\mathbb N$ by the space of configurations -- discrete Radon measures $\gamma=\sum_i\delta_{ x_i}$ on $X$, where $\delta_{ x_i}$ is the Dirac measure with mass at $x_i$.The spatial falling factorials $(\gamma)_n:=\sum_{ i_1}\sum_{ i_2\ne i_1}\dotsm\sum_{ i_n\ne i_1,\dots, i_n\ne i_{ n-1}}\delta_{ (x_{ i_1},x_{ i_2},\dots,x_{ i_n})}$ can be naturally extended to mappings $M^{ (1)}(X)\ni\omega\mapsto (\omega)_n\in M^{ (n)}(X)$, where $M^{ (n)}(X)$ denotes the space of $\mathbb F$-valued, symmetric (for $n\ge2$) Radon measures on $X^n$. There is a natural duality between $M^{ (n)}(X)$ and the space $\mathcal { CF}^{ (n)}(X)$ of $\mathbb F$-valued, symmetric continuous functions on $X^n$ with compact support. The Stirling operators of the first and second kind, $\mathbf{ s}(n,k)$ and $\mathbf{ S}(n,k)$, are linear operators, acting between spaces $\mathcal { CF}^{ (n)}(X)$ and $\mathcal { CF}^{ (k)}(X)$ such that their dual operators, acting from $M^{ (k)}(X)$ into $M^{ (n)}(X)$, satisfy $(\omega)_n=\sum_{ k=1}^n\mathbf{ s}(n,k)^*\omega^{ \otimes k}$ and $\omega^{ \otimes n}=\sum_{ k=1}^n\mathbf{ S}(n,k)^*(\omega)_k$, respectively. We derive combinatorial properties of the Stirling operators, present their connections with a generalization of the Poisson point process and with the Wick ordering under the canonical commutation relations.
Categories: math.CO math-ph math.FA math.MP math.PR