Heterogeneous Stirling numbers and heterogeneous Bell polynomials
Taekyun Kim, Dae San Kim
TL;DR
This work introduces a parametric family of heterogeneous Stirling numbers $H_{\lambda}(n,k)$ that interpolate between the classical Stirling numbers ${n \brace k}$ ($\lambda\to 0$) and Lah numbers $L(n,k)$ ($\lambda\to 1$), along with the heterogeneous first-kind numbers $G_{\lambda}(n,k)$ and the heterogeneous Bell polynomials $H_{n,\lambda}(x)$. It develops generating functions, explicit formulas, and recurrences for these objects, and extends the construction to heterogeneous $r$-Stirling numbers $H_{\lambda}^{(r)}(n+r,k+r)$ and their corresponding $r$-Bell polynomials $H_{n,\lambda}^{(r)}(x)$, including a Dobinski-like representation for the Bell polynomials. The framework leverages degenerate exponentials and factorials to unify and generalize classical combinatorial sequences, yielding rich identities such as $H_{\lambda}(n,k)=\sum_{l=k}^{n}{l \brace k}{n \brack l}\lambda^{n-l}$ and associated recurrences. These results open avenues for deeper combinatorial interpretation, connections to other special functions, and potential applications in related mathematical and applied domains.
Abstract
This paper introduces a novel generalization of Stirling and Lah numbers, termed ``heterogeneous Stirling numbers," which smoothly interpolate between these classical combinatorial sequences. Specifically, we define heterogeneous Stirling numbers of the second and first kinds, demonstrating their convergence to standard Stirling numbers for lambda=0 and to (signed) Lah numbers for lambda =1. We derive fundamental properties, including generating functions, explicit formulas, and recurrence relations. Furthermore, we extend these concepts to heterogeneous Bell polynomials, obtaining analogous results such as generating function, combinatorial identity and Dobinski-like formula. Finally, we introduce and analyse heterogeneous r-Stirling numbers of the second kind and their associated r-Bell polynomials.