Stirling operators in spatial combinatorics

Abstract

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 $(z{)}_n=z(z-1)\cdots (z-n+1)$ 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 $z^k$, $k\le n$ 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 $(\gamma {)}_n:={\sum }_{i_1}{\sum }_{i_2\ne i_1}\cdots {\sum }_{i_n\ne i_1,\dots ,i_n\ne i_{n-1}}{\delta }_{(x_{i_1},x_{i_2},\dots ,x_{i_n})}$ 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 $n\ge 2$) Radon measures on $X^n$. There is a natural duality between $M^{(n)}(X)$ and the space ${\mathcal{C}\mathcal{F}}^{(n)}(X)$ 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 ${\mathcal{C}\mathcal{F}}^{(n)}(X)$ and ${\mathcal{C}\mathcal{F}}^{(k)}(X)$ such that their dual operators, acting from $M^{(k)}(X)$ into $M^{(n)}(X)$, satisfy $(\omega {)}_n={\sum }_{k=1}^n\mathbf{s}(n,k{)}^{\ast }{\omega }^{\otimes k}$ and ${\omega }^{\otimes n}={\sum }_{k=1}^n\mathbf{S}(n,k{)}^{\ast }(\omega {)}_k$, 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.

References in corpus (1)

• On a spectral representation for correlation measures in configuration space analysis

Cited by in corpus (1)

• A biorthogonal approach to the infinite dimensional fractional Poisson measure