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.

通过应用p-adic积分，在Simsek中（Montes Taurus J Pure Appl Math 3（1）：38–61，2021年），我们为涉及新的组合和和数的特殊数和多项式构造了生成函数。本文的目的是使用这些组合和和数来推导与Bernstein基函数，Fibonacci数，Harmonic数，交替Harmonic数，高阶Bernoulli多项式，二项式系数和黎曼积分的新积分公式。我们还研究涉及这些数字的未解决问题。此外，我们给出了这些数字，Digamma函数和Euler常数之间的关系。此外，通过应用这些组合和的特殊值，我们给出了多个Hurwitz zeta函数的分解，该函数对高阶Bernoulli多项式进行插值。最后，我们通过一些评论和观察给出本文结果的结论。