Küçükoğlu, İremŞimşek, Yılmaz2021-02-192021-02-1920180354-5180https://doi.org/10.2298/FIL1820879Khttps://hdl.handle.net/20.500.12868/510International Conference on Approximation and Computation - Theory and Applications (ACTA) -- NOV 30-DEC 02, 2017 -- Belgrade, SERBIAKUCUKOGLU, IREM/0000-0001-9100-2252The 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 Apostol-Bernoulli 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.eninfo:eu-repo/semantics/openAccessGenerating functionsFunctional equationsPartial differential equationsStirling numbers of the second kindEuler numbers of the second kindApostol-Euler type polynomials of the second kindlambda-Bernoulli numbersBell numbersCombinatorial sumsBinomial coefficientsArithmetical functionsConference Object10.2298/FIL1820879K322068796891Q3WOS:000463447500004N/A