Study on Two New Numbers and Polynomials Numbers and Polynomials Arising from the Fermionic p-adic Integral on \(\mathbb{Z}\)p

Abstract

p-adic analysis and their applications is used p-adic distributions, p-adic measure, p-adic integrals, p-adic L-function and other generalized functions. In addition, among the many ways to investigate and construct generating functions for special polynomials and numbers, one of the most important techniques is the p-adic Fermionic integral over \(\mathbb{Z}\)p. In this paper, we introduce new numbers and polynomials arising from the Fermionic p-adic integral on \(\mathbb{Z}\)p. First, we introduce new numbers and polynomials as one of generalizations of Changhee numbers and polynomials of order r (r \(\epsilon\) \(\mathbb{N}\)), which are called the generalized Changhee numbers and polynomials. We explore some interesting identities and explicit formulas of these numbers and polynomials. Second, we define new numbers and polynomials as one of generalizations of Catalan numbers and polynomials of order r (r \(\epsilon\) \(\mathbb{N}\)), which are called the generalized Catalan numbers and polynomials. We also study some combinatorial identities and explicit formulas of these numbers and polynomials.