© 2018 Jangjeon Mathematical Society. All rights reserved.The main motivation of this paper is to study and investigate a new family of combinatorial numbers with their generating functions. Firstly, we obtain some finite series representations including well-known numbers such as the Apostol-Bernoulli numbers, the Apostol-Euler numbers, a family of combinatorial numbers, the Daehee numbers, the Changhee numbers and the Stirling numbers of the second kind. Secondly, applying Mellin transform to these functions, we give interpolation functions for these numbers. We investigate some properties of these functions and other related complex valued functions. We observe that some special values of these functions give us the terms of some well-known infinite series. Thus, these functions unify the terms of some well-known identities and functions such as Hasse identity, the polylogarithm function, the digamma function, the Riemann zeta functions, the alternating Riemann zeta function, the Hurwitz zeta function, the alternating Hurwitz zeta function, the Hurwitz-Lerch zeta function and the other functions. Moreover, we give some remarks and observations about these functions related to some special numbers and polynomials such as the Stirling numbers of the second kind, the harmonic numbers, the array polynomials and also related to hypergeometric functions, the family of zeta functions. We also give not only Riemann integral representation, but also Cauchy integral representations for this new family of combinatorial numbers. Finally, in order to compute numerical values of these interpolation functions and other related complex valued functions, we present two algorithms. Furthermore, by using these algorithms, we provide some plots of these functions. Also, we investigate the effects of their parameters.