Generalized multipliers for a left-invertible operator T, whose formal Laurent series , actually represent analytic functions on an annulus or a disc are investigated. We show that they are coefficients of analytic functions and characterize the commutant of some left-invertible operators, which satisfies certain conditions in its terms. In addition, we prove that the set of multiplication operators associated with a weighted shift on a rootless directed tree lies in the closure of polynomials in z and 1z of the weighted shift in the topologies of strong and weak operator convergence.