Explicit formulas for p-adic integrals: approach to p-adic distributions and some families of special numbers and polynomials

Abstract: The main objective of this article is to give and classify new formulas of $p$-adic integrals and blend these formulas with previously well known formulas. Therefore, this article gives briefly the formulas of $p$-adic integrals which were found previously, as well as applying the integral equations to the generating functions and other special functions, giving proofs of the new interesting and novel formulas. The $p$-adic integral formulas provided in this article contain several important well-known families of special numbers and special polynomials such as the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers, the Lah numbers, the Peters numbers and polynomials, the central factorial numbers, the Daehee numbers and polynomials, the Changhee numbers and polynomials, the Harmonic numbers, the Fubini numbers, combinatorial numbers and sums. In addition, we defined two new sequences containing the Bernoulli numbers and Euler numbers. These two sequences include central factorial numbers, Bernoulli numbers and Euler numbers. Some computation formulas and identities for these sequences are given. Finally we give further remarks, observations and comments related to content of this paper.