First exit time from a bounded interval for pseudo-processes driven by the equation $\mathrm{\partial }\mathrm{/}\mathrm{\partial }t=(-1{)}^{N-1}{\mathrm{\partial }}^{2N}\mathrm{/}\mathrm{\partial }{x}^{2N}$

Let $N$ be a positive integer. We consider pseudo-Brownian motion $X=(X(t){)}_{t\ge 0}$ driven by the high-order heat-type equation $\mathrm{\partial }\mathrm{/}\mathrm{\partial }t=(-1{)}^{N-1}{\mathrm{\partial }}^{2N}\mathrm{/}\mathrm{\partial }{x}^{2N}$. Let us introduce the first exit time {\tau}ab from a bounded interval $(a,b)$ by $X$ ($a,b\in \mathbb{R}$). In this paper, we provide a representation of the joint pseudo-distribution of the vector $({\tau }_{ab},X({\tau }_{ab}))$ by means of Vandermonde-like determinants. The method we use is based on the Feynman-Kac functional related to pseudo-Brownian motion which leads to a boundary value problem. In particular, the pseudo-distribution of the location of $X$ at time ${\tau }_{ab}$, namely $X({\tau }_{ab})$, admits a fine expression involving famous Hermite interpolating polynomials.