Degenerate weighted Stirling numbers

Abstract

We define the degenerate weighted Stirling numbers of the first and second kinds, S 1( n, k, λ ∥ θ) and S( n, k, λ ∥ θ). By specializing λ and θ we can obtain the Stirling numbers, the weighted Stirling numbers and the degenerate Stirling numbers. Basic properties of S 1( n, k, λ ∥ θ) and S( n, k, λ ∥ θ), such as recurrence formulas and combinatorial interpretations, are presented, and a theorem which relates S 1( n, k, λ ∥ θ) and S( n, k, λ ∥ θ) to each other, and to other special numbers, is proved. This theorem provides a unified approach to a number of special cases which have recently appeared in the literature.