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The first aim of this paper is to give identities and relations for a new family of the combinatorial numbers and the Apostol-Euler type numbers of the second kind, the Stirling numbers, the ApostolBernoulli type numbers, the Bell numbers and the numbers of the Lyndon words by using some techniques including generating functions, functional equations and inversion formulas. The second aim is to derive some derivative formulas and combinatorial sums by applying derivative operators including the Caputo fractional derivative operators. Moreover, we give a recurrence relation for the Apostol-Euler type numbers of the second kind. By using this recurrence relation, we construct a computation algorithm for these numbers. In addition, we derive some novel formulas including the Stirling numbers and other special numbers. Finally, we also some remarks, comments and observations related to our results.

Identities and derivative formulas for the combinatorial and Apostol-Euler type numbers by their generating functions