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 ∈ N to the falling factorials (z) n = z(z−1) • • • (z−n+1) of an argument z from F = R or C, 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 N by the space of configurations-discrete Radon measures γ = i δ x i on X, where δ x i is the Dirac measure with mass at x i . The spatial falling factorials (γ, 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 X n with 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 (ω) n = n k=1 s(n, k) * ω ⊗k and ω ⊗n = n k=1 S(n, k) * (ω) 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.

Intertwining and Duality for Consistent Markov Processes

“…These relations are similar to relations between moments and factorial moments of integer-valued random variables with Stirling numbers, see [14,Chapter 5]. A systematic treatment in terms of Stirling operators is found in [22].…”

“…Equation (3.4) will be used in Section 4 below to recover known self-duality functions for particle systems on finite set from the abstract Theorem 3.5. We refer to [22] for further properties of the generalized falling factorial polynomials. Our first main result is an intertwining relation between the Markov semigroup (P t ) t≥0 and (p…”

Philosophy of Natural Numbers

“…To be friendly to more wide audience, we restrict out explanations to descriptions of main constructions and formulation of some particular results. For detailed discussions and extended references we refer to the recent paper [2].…”