A new class of the entire function of order one: a case study

TL;DR

This paper introduces a new class of entire functions of order one with positive coefficients and complex zeros, establishing properties of zeros on the critical line and providing new product representations, including for hyperbolic cosine.

Contribution

It is the first investigation of this new class of entire functions, with theorems linking zeros to the critical line and new product formulas for hyperbolic functions.

Findings

Zeros of the functions lie on the critical line.

Established an equivalent representation theorem for the critical line.

Discovered new product formulas for hyperbolic cosine and sinc functions.

Abstract

In this article, a new class of the entire function of order one, expressed by the series and product representations with the real positive coefficients and complex zeros, is investigated for the first time. The entire function on the critical line deduces an even entire function of order one. It is proved that the real part of the complex zeros is equal to the critical line. An equivalent representation theorem is obtained to set up the sufficient conditions for the critical line for the entire function. As a typical example, the critical line for the special hyperbolic cosine function obtained by the present theorem agrees with the result of Euler. We also discover the new products of the hyperbolic cosine and sinc functions.