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### Stirling operators in spatial combinatorics / Dmitri Finkelshtein, Yuri Kondratiev, Eugene Lytvynov, Maria João Oliveira

Journal of Functional Analysis, Volume: 282, Issue: 2, Start page: 109285

Swansea University Authors:

• Accepted Manuscript under embargo until: 19th October 2022

DOI (Published version): 10.1016/j.jfa.2021.109285

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 $(z)_n=z(z-1)\dotsm (z-n+1)$ of an argument $z$ from $\mathbb F=\mathbb R\te... Full description Published in: Journal of Functional Analysis 0022-1236 Elsevier BV 2022 https://cronfa.swan.ac.uk/Record/cronfa58358 No Tags, Be the first to tag this record! Abstract: We define and study a spatial (infinite-dimensional) counterpart of Stirling numbers. In classical combinatorics, the Pochhammer symbol$(m)_n$can be extended from a natural number$m\in\mathbb N$to the falling factorials$(z)_n=z(z-1)\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$, respectively. In the case where$X\$ has only a single point, the Stirling operators can be identified with Stirling numbers. 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. Spatial falling factorials; Stirling operators; Poisson functional; Wick ordering for canonical commutation relations College of Science 2 109285