  Recently, the numbers $Y_{n}(\lambda )$ and the polynomials
$Y_{n}(x,\lambda)$ have been introduced by the second author [22]. The purpose
of this paper is to construct higher-order of these numbers and polynomials
with their generating functions. By using these generating functions with their
functional equations and derivative equations, we derive various identities and
relations including two recurrence relations, Vandermonde type convolution
formula, combinatorial sums, the Bernstein basis functions, and also some well
known families of special numbers and their interpolation functions such as the
Apostol--Bernoulli numbers, the Apostol--Euler numbers, the Stirling numbers of
the first kind, and the zeta type function. Finally, by using Stirling's
approximation for factorials, we investigate some approximation values of the
special case of the numbers $Y_{n}\left( \lambda \right) $.
