A new class of the entire functions: a study of two cases from Euler

In this article an alternative infinite product for a special class of the entire functions are studied by using some results of the Laguerre-P\'{o}lya entire functions. The zeros for a class of the special even entire functions are discussed in detail. It is proved that the infinite product and series representations for the hyperbolic and trigonometric cosine functions, which are coming from Euler, are our special cases.Comment: 10 page

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

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.Comment: V1. there are 11 pages. V2, there are 20 pages. We consider some new contents for the entire function