Spectral analysis of two doubly infinite Jacobi matrices with exponential entries

Abstract We provide a complete spectral analysis of all self-adjoint operators acting on l 2 ( 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 + α q − n δ m , n + δ m , n + 1 , respectively, where q ∈ ( 0 , 1 ) and α ∈ R . As an application, we derive orthogonality relations for the Ramanujan entire function and the third Jackson q -Bessel function.