  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
$\partial/\partial t=(-1)^{N-1}\partial^{2N}/\partial x^{2N}$. 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.
