In this paper, we propose an approach that utilizes the generalized Fibonacci sequence as a tool for studying the infinite Hankel matrix of finite rank, associated with a nonzero sequence \(S = \{s_n\}_ {n\ge 0}\) of real numbers. We offer detailed characterizations of both the generating and representing measures for any sequence associated with a Hankel matrix of finite rank. Furthermore, we make use of established results on the Z-transform of generalized Fibonacci sequences to investigate the connection between infinite Hankel matrices of finite rank and the Hamburger moment problem for such sequences. Indeed, we present some rephrased findings and newly derived results. Additionally, illustrative examples are provided to demonstrate the efficacy of our approach.
