Authors: İREM KÜÇÜKOĞLU, YILMAZ ŞİMŞEK

Abstract: The main purpose of this paper is to obtain computational identities and formulas for a certain class of combinatorial-type numbers and polynomials. By the aid of the generating function technique, we derive a recurrence relation and an infinite series involving the aforementioned class of combinatorial-type numbers. By applying the Riemann integral to the combinatorial-type polynomials with multivariables, we present some integral formulas for these polynomials, including the Bernoulli numbers of the second kind. By the implementation of the $p$-adic integral approach to the combinatorial-type polynomials with multivariables, we also obtain formulas for the Volkenborn integral and the fermionic $p$-adic integral of these polynomials. Furthermore, we provide an approximation for the combinatorial-type numbers with the aid of the Stirling's approximation for factorials. By coding some of our results in Mathematica using the Wolfram programming language, we also provide some numerical evaluations and illustrations on the combinatorial-type numbers and their Stirling's approximation with table and figures. We also give some remarks and observations on the combinatorial-type numbers together with their relationships to other well-known special numbers and polynomials. As a result of these observations, we derive some computation formulas containing the Dirichlet series involving the M\"{o}bius function, the Bernoulli numbers, the Catalan numbers, the Stirling numbers, the Apostol--Bernoulli numbers, the Apostol--Euler numbers, the Apostol--Genocchi numbers and some kinds of combinatorial numbers. Besides, some inequalities for the combinatorial-type numbers are presented. Finally, we conclude this paper by briefly overviewing the results with their potential applications.

Keywords: Generating functions, $p$-adic integrals, Catalan numbers, special numbers and polynomials, Combinatorial numbers, Stirling approximation

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