Some identities of special numbers and polynomials arising from p$-adic integrals on Zp

TL;DR

This paper introduces new degenerate and type 2 versions of Bernoulli and Euler polynomials and numbers, deriving identities and relations using p-adic integrals on Zp.

Contribution

It presents novel degenerate and type 2 Bernoulli and Euler polynomials and numbers, along with their identities and distribution relations, using p-adic integral techniques.

Findings

Derived identities and distribution relations for the new polynomials.

Established analogues of Bernoulli's interpretation of powers.

Utilized bosonic and fermionic p-adic integrals on Zp.

Abstract

In recent years, studying degenerate versions of various special polynomials and numbers have attracted many mathematicians. Here we introduce degenerate type 2 Bernoulli polynomials, fully degenerate type 2 Bernoulli polynomials and degenerate type 2 Euler polynomials, and their corresponding numbers, as degenerate and type 2 versions of Bernoulli and Euler numbers. Regarding to those polynomials and numbers, we derive some identities, distribution relations, Witt type formulas and analogues for the Bernoulli's interpretation of powers of the first $m$ positive integers in terms of Bernoulli polynomials. The present study was done by using the bosonic and fermionic $p$-adic integrals on $\mathbb{Z}_p$.