In this paper, we mainly concerned with an alternate form of the generating
functions for a certain class of combinatorial numbers and polynomials. We
give matrix representations for these numbers and polynomials with their applications. We also derive various identities such as Rodrigues-type
formula, recurrence relation and derivative formula for the aforementioned
combinatorial numbers. Besides, we present some plots of the generating
functions for these numbers. Furthermore, we give relationships of these
combinatorial numbers and polynomials with not only Bernstein basis
functions, but the two-variable Hermite polynomials and the number of cyclic
derangements. We also present some applications of these relationships. By
applying Laplace transform and Mellin transform respectively to the
aforementioned functions, we give not only an infinite series representation,
but also an interpolation function of these combinatorial numbers. We also
provide a contour integral representation of these combinatorial numbers. In
addition, we construct exponential generating functions for a new family of
numbers arising from the linear combination of the numbers of cyclic
derangements in the wreath product of the finite cyclic group and the
symmetric group of permutations of a set. Finally, we analyse the
aforementioned functions in probabilistic and asymptotic manners, and we
give some of their relationships with not only the Laplace distribution, but
also the standard normal distribution. Then, we provide an asymptotic power
series representation of the aforementioned exponential generating
functions.