  In this paper, we present a framework for the recursion method applied within
the Liouvillian formalism, enabling the computation of response functions for a
wide range of quantum operators. Indeed, unlike most previous literature on the
Liouvillian recursion method, this framework readily computes electronic
Green's function. We demonstrate the method by showcasing its application to a
temperature dependence study of a $2\times2$ Hubbard model. Our results show
that the recursion method accurately computes response functions and provides
valuable insights into quantum many-body systems. While the method has already
been successfully applied to study of condensed matter, further efforts are
required to develop efficient algorithms for larger and more realistic systems.
Tensor network methods and other variational techniques hold promise for
tackling these challenges. Overall, the recursion method offers a flexible
approach for investigating strongly correlated system.
