Abstract
The nth r-extended Lah–Bell number is defined as the number of ways a set with n+r elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to introduce incomplete r-extended Lah–Bell polynomials and complete r-extended Lah–Bell polynomials respectively as multivariate versions of r-Lah numbers and the r-extended Lah–Bell numbers and to investigate some properties and identities for these polynomials. From these investigations we obtain some expressions for the r-Lah numbers and the r-extended Lah–Bell numbers as finite sums.
Highlights
It is well known that the unsigned Lah number L(n, k) (n ≥ k ≥ 0) counts the number of ways a set with n elements can be partitioned into k nonempty linearly ordered subsets
3 Conclusion There are various methods of studying special numbers and polynomials, for example, generating functions, combinatorial methods, umbral calculus, p-adic analysis, differential equations, probability theory, orthogonal polynomials, and special functions. These ways of investigating special polynomials and numbers can be applied to degenerate versions of such polynomials and numbers
In recent years, many mathematicians have drawn their attention to studies of degenerate versions of many special polynomials and numbers by using the aforementioned means ([9, 10, 14] and references therein)
Summary
It is well known that the unsigned Lah number L(n, k) (n ≥ k ≥ 0) counts the number of ways a set with n elements can be partitioned into k nonempty linearly ordered subsets (see [4, 7, 8]). The r-extended Lah–Bell number BLn,r is defined as the number of ways a set with n + r elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks (see [8]). From (4) we see that the generating function of r-extended Lah-Bell numbers is given by t e 1–t. Like Stirling numbers of both kinds, Lah numbers, and idempotent numbers, appear in many combinatorial and number-theoretic identities involving complete and incomplete Bell polynomials. The aim of this paper is to introduce the incomplete r-extended Lah-Bell polynomials and the complete r-extended Lah-Bell polynomials and to investigate some properties and identities for these polynomials From these investigations we obtain some expressions for the r-Lah numbers and the r-extended Lah–Bell numbers as finite sums
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