Application of the Euler Operator to Generating Functions of Combinatorial Simsek Numbers

This paper aims to investigate not only combinatorial Simsek numbers of the sixth kind together with their generating functions, but also solutions of the open problems, given in [1] associated with ordinary generating functions for these numbers. Using Euler operator, a partial solution for one of the open problems, posed by Simsek [1], are obtained. Many relations among these numbers, Legendre polynomials, Chebyshev polynomials and Stirling numbers are given. Moreover, using these functions, special values and formulas are derived. Some special cases with their numerical values of the results of this paper are given.