Heterogeneous Stirling Numbers and Heterogeneous Bell Polynomials

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 as \(\lambda \rightarrow 0\) and to (signed) Lah numbers as \(\lambda \rightarrow 1\). 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.

DOI 10.1134/S1061920825601065