Abstract
Degenerate versions of special polynomials and numbers applied to social problems, physics, and applied mathematics have been studied variously in recent years. Moreover, the (s-)Lah numbers have many other interesting applications in analysis and combinatorics. In this paper, we divide two parts. We first introduce new types of both degenerate incomplete and complete s-Bell polynomials respectively and investigate some properties of them respectively. Second, we introduce the degenerate versions of complete and incomplete Lah-Bell polynomials as multivariate forms for a new type of degenerate s-extended Lah-Bell polynomials and numbers respectively. We investigate relations between these polynomials and degenerate incomplete and complete s-Bell polynomials, and derive explicit formulas for these polynomials.
Highlights
For nonnegative integers n, k, s such that n ≥ k, the s-Lah number Ls(n, k) counts the number of partitions of a set with n + s elements into k + s ordered blocks such that s distinguished elements have to be in distinct ordered blocks [1–5]
In [7], as an example considering the psychological burden of baseball hitters, it well expresses the starting point of degenerate special polynomials and numbers being studied by many scholars
We study several properties and explicit formulas for them
Summary
K, s such that n ≥ k, the s-Lah number Ls(n, k) counts the number of partitions of a set with n + s elements into k + s ordered blocks such that s distinguished elements have to be in distinct ordered blocks [1–5]. In [7], as an example considering the psychological burden of baseball hitters, it well expresses the starting point of degenerate special polynomials and numbers being studied by many scholars Both the complete and incomplete Bell polynomials are multivariate forms for Bell polynomials and Stirling numbers of the second kind, respectively. We investigate explicit formulas for degenerate complete and incomplete s-extended Lah-Bell polynomials, respectively. For a nonnegative integer s, the s-Stirling numbers S2(s)(n, k) of the second kind are given by the generating function. It is well known that [2] an explicit formula and the generating function of s-Lah Bell Ls(n, k) are given by, respectively. Kim et al [2] introduced the s-extended Lah-Bell polynomials Lbn,s(x) given by the generating function xt.
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