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 $q^{-n+1}δ_{m,n-1}+q^{-n}δ_{m,n+1}$ and $δ_{m,n-1}+αq^{-n}δ_{m,n}+δ_{m,n+1},$ respectively, where $q\in(0,1)$ and $α\in\mathbb{R}$. As an application, we derive orthogonality relations for the Ramanujan entire function and the third Jackson $q$-Bessel function.

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