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

Combinatorial identities associated with new families of the numbers and polynomials and their approximation values