On All Real Zeros for a Class of Even Entire Functions

The present paper deals with a class of even entire functions of order $\rho =1$ and genus $\vartheta =0$ of the polynomials form, \[ \sum\limits_{m=0}^\infty {\frac{\left( {-1} \right)^m\Phi ^{\left( {2m} \right)}\left( 0 \right)}{\Gamma \left( {2m+1} \right)}x^{2m}} =\Phi \left( 0 \right)\prod\limits_{k=1}^\infty {\left( {1-\frac{x}{\ell _k }} \right)} , \] where $\Phi\left( 0 \right)\ne 0$, real numbers $x$, nonnegative integers $m$, and $\ell _k \ne 0$ are all of the nonzero roots with $\sum\limits_{k=1}^\infty {1/\left| {\ell _k } \right|} <\infty $ and natural numbers $k$. We provide an efficient criterion for the polynomials with only real zeros. We also prove that the conjecture of Jensen is our special case.