# Article ID: MTJPAM-D-19-00005

## Title: Explicit Formulas for p-adic Integral: Approach to p-adic Distributions and Some Families of Special Numbers and Polynomials

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-19-00005; Volume 1 / Issue 1 / Year 2019, Pages 1-76

Document Type: Research Paper

Author(s): Yilmaz Simsek a

aDepartment of Mathematics, Faculty of Science University of Akdeniz TR-07058 Antalya-Turkey

Received: 2 September 2019, Accepted: 9 November 2019, Available online: 28 November 2019.

Corresponding Author: Yilmaz Simsek (Email address: ysimsek@akdeniz.edu.tr)

Full Text: PDF

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 of the first kind and the Euler numbers of the first kind. These two sequences include central factorial numbers, Bernoulli numbers of the first kind and the Euler numbers of the first kind. Some computation formulas and identities for these sequences are given. Finally, we give further remarks, observations and comments related to content of this paper.

Keywords: p-adic q-integrals, Volkenborn integral, Generating function, Special functions, Bernoulli numbers and polynomials of the first kind, Euler numbers and polynomials of the first kind, Stirling numbers, Lah numbers, Peters numbers and polynomials, Central factorial numbers, Daehee numbers and polynomials, Changhee numbers and polynomials, Harmonic numbers, Fubini numbers, Combinatorial numbers and sums

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