PUB - Publikationen an der Universität Bielefeld

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 epsilon N to the falling factorials (z)(n)= z(z - 1) . . . (z - n + 1) of an argument zfrom F = RorC, and Stirling numbers of the first and second kinds are the coefficients of the expansions of (z)(n) through z(k), k <= n and vice versa. When taking into account spatial positions of elements in a locally compact Polish space X, we replace Nby the space of configurations-discrete Radon measures gamma= Sigma(i) delta(xi) on X, where delta(xi) is the Dirac measure with mass at x(i). The spatial falling factorials (gamma)(n):= Sigma (i1) Sigma(i2 not equal i1) . .. , Sigma(in not equal i1),...,(delta)(in not equal in-1)(x(i1),x(i2),...,x(in)) can be naturally extended to mappings M-(1)(X) (SIC) omega -> (omega)(n) epsilon M-(n)(X), where M-(n)(X) denotes the space of F-valued, symmetric (for n >= 2) Radon measures on X-n. There is a natural duality between M-(n)(X) and the space CF(n)(X) of F-valued, symmetric continuous functions on Xnwith compact support. The Stirling operators of the first and second kind, s(n, k) and S(n, k), are linear operators, acting between spaces CF(n)(X) and CF(k)(X) such that their dual operators, acting from M-(k)(X) into M-(n)(X), satisfy (omega)(n)= Sigma(k=1s)(n)(n, k)(*)omega(circle times k) and omega(circle times n) = Sigma S-k= 1(n)( n, k)(*)(omega)(k), respectively. In the case where Xhas 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. (C) 2021 Elsevier Inc. All rights reserved.