New integral formulas and identities involving special numbers and functions derived from certain class of special combinatorial sums

TL;DR

This paper introduces new integral formulas and identities involving special numbers and functions derived from combinatorial sums, with applications to various classical mathematical entities and open problems.

Contribution

It constructs generating functions for special numbers using p-adic integrals and derives novel identities involving special functions, numbers, and combinatorial sums.

Findings

Derived new identities for Bernstein basis functions and Fibonacci numbers.

Established relations between special numbers y(n,λ), the Digamma function, and Euler's constant.

Presented new integral formulas for the Riemann integral.

Abstract

By applying p-adic integral on the set of p-adic integers in [27] (Interpolation Functions for New Classes Special Numbers and Polynomials via Applications of p-adic Integrals and Derivative Operator, Montes Taurus J. Pure Appl. Math. 3 (1), ...--..., 2021 Article ID: MTJPAM-D-20-00000), we constructed generating function for the special numbers and polynomials involving the following combinatorial sum and numbers: y(n,\lambda )=\sum_{j=0}^{n}\frac{(-1)^{n}}{(j+1)\lambda ^{j+1}\left(\lambda -1\right) ^{n+1-j}} The aim of this paper is to use the numbers y(n,{\lambda}) to derive some new and novel identities and formulas associated with the Bernstein basis functions, the Fibonacci numbers, the Harmonic numbers, the alternating Harmonic numbers, binomial coefficients and new integral formulas for the Riemann integral. We also investigate and study on open problems involving the numbers y(n,{\lambda}) in [27]. Moreover, we give relation among the numbers y(n,(1/2)), the Digamma function, and the Euler constant. Finally, we give conclusions for the results of this paper with some comments and observations.