By applying p-adic integral, in Simsek (Montes Taurus J Pure Appl Math 3(1):38-61, 2021), we constructed generating function for the special numbers and polynomials involving novel combinatorial sums and numbers. The aim of this paper is to use these combinatorial sums and numbers to derive various new formulas and relations associated with the Bernstein basis functions, the Fibonacci numbers, the Harmonic numbers, the alternating Harmonic numbers, the Bernoulli polynomials of higher order, binomial coefficients and new integral formulas for the Riemann integral. We also investigate and study on open problems involving these numbers. Moreover, we give relation among these numbers, the Digamma function, and the Euler constant. Moreover, by applying special values of these combinatorial sums, we give decomposition of the multiple Hurwitz zeta function which interpolates the Bernoulli polynomials of higher order. Finally, we give conclusions for the results of this paper with some comments and observations.