In this article, we are study the question of existence of integral equation for the monic $\mathcal{H}$ermite polynomials ${H}_{n}$, where the intervening real function does not depend on the index $n$, well-known by the linear functional $\mathscr{W}_{x}$ given by its moments ${H}_{n}(x)=\left⟨\mathscr{W}_{x},t^{n}\right⟩$, $n\geq 0$, $| x| <\infty$. Also, we obtain some properties of the zeros of this intervening function. Furthermore, we obtain an integral representation of the Dirac mass $\delta _{x},$ for every real number $x$.