  We provide a complete spectral analysis of all self-adjoint operators acting
on $\ell^{2}(\mathbb{Z})$ which are associated with two doubly infinite Jacobi
matrices with entries given by $$ q^{-n+1}\delta_{m,n-1}+q^{-n}\delta_{m,n+1}
$$ and $$ \delta_{m,n-1}+\alpha q^{-n}\delta_{m,n}+\delta_{m,n+1}, $$
respectively, where $q\in(0,1)$ and $\alpha\in\mathbb{R}$. As an application,
we derive orthogonality relations for the Ramanujan entire function and the
third Jackson $q$-Bessel function.
