FORMULAS AND COMBINATORIAL IDENTITIES FOR CATALAN-TYPE NUMBERS AND POLYNOMIALS: THEIR ANALYSIS WITH COMPUTATIONAL ALGORITHMS

In this paper, we aim to provide generating functions for a higher-order expansion of a certain class of Catalan-type numbers and polynomials, and to give some computational algorithms for evaluating these numbers and polynomials. With the implementation of these computational algorithms in Mathematica by Wolfram programming language, we provide some plots drawn depending on varying special cases of the Catalan-type polynomials of higher-order. By using generating functions, we also derive some formulas and combinatorial identities. By applying not only the Riemann integral, but also the p-adic integrals to these formulas, we get some integral formulas involving the Catalan-type numbers and polynomials, the factorial polynomials, the Stirling numbers, the Bernoulli numbers of the second kind, the Daehee and Changhee numbers and polynomials. By using these integral formulas, we derive other combinatorial sums including the Catalan-type numbers. In addition, we provide some finite and infinite series representations which arise from the Catalan-type numbers.