Spectral analysis of two doubly infinite Jacobi matrices with exponential entries

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.