It is shown that the sequence of the generalized Bell polynomials S_n(x) is convex under some restrictions of the parameters involved. A kind of recurrence relation for S_n(x) is established, and some numbers related to the generalized Bell numbers and their properties are investigated.

In this paper we define generalized exponential polynomials by means of the generalization of the Mellin derivative $\left(xD\right)$. We give different proofs for some known results and obtain a new recurrence relation and a new operator formula for generalized exponential polynomials. We also define generalized geometric polynomials by means of the generalization of the Mellin derivative $\left(xD\right)$ and obtain their basic properties.

Ther-Whitney numbers of the second kind are a generalization of all the Stirling-type numbers of the second kind which are in line with the unified generalization of Hsu and Shuie. In this paper, asymptotic formulas forr-Whitney numbers of the second kind with integer and real parameters are obtained and the range of validity of each formula is established.

A barred preferential arrangement is a preferential arrangement, onto which in-between the blocks of the preferential arrangement a number of identical bars are inserted. We offer a generalisation of barred preferential arrangements by making use of the generalised Stirling numbers proposed by Hsu and Shiue (1998). We discuss how these generalised barred preferential arrangements offer a unified combinatorial interpretation of geometric polynomials. We also discuss asymptotic properties of these numbers. (C) 2019 The Authors. Published by Elsevier B.V.