Note on the Euler Numbers and Polynomials

TL;DR

This paper explores the properties of Euler functions using Fourier transforms, deriving formulas related to infinite series and establishing identities between Euler numbers and Stirling numbers.

Contribution

It introduces new identities linking Euler numbers with Stirling numbers and employs Fourier analysis to derive formulas for Euler functions.

Findings

Derived formulas for Euler functions via Fourier transform.

Established identities between Euler numbers and Stirling numbers.

Presented new series representations involving Euler functions.

Abstract

In this paper we investigate the properties of the Euler functions. By using the Fourier transform for the Euler function, we derive the interesting formula related to the infinite series. Finally we give some interesting identities between the Euler numbers and the second kind stirling numbers.