  In this paper we demonstrate a connection between the roots of a certain
sequence of orthogonal polynomials on the real line and the linear instability
of a $x$-directionally homogeneous background velocity profile $u^b(x,y) =
\cos(y)$ in the quasi-geostrophic shallow water (QG) equation in a domain with
periodic boundaries in the $y$-direction. Using the relationship we establish,
we then prove that there exists a unique unstable mode for each horizontal wave
number $0<k<1$ and provide mathematically rigorous estimates of the associated
growth rate.
