  We construct a new family of strong linearizations of rational matrices
considering the polynomial part of them expressed in a basis that satisfies a
three term recurrence relation. For this purpose, we combine the theory
developed by Amparan et al., MIMS EPrint 2016.51, and the new linearizations of
polynomial matrices introduced by Fa{\ss}bender and Saltenberger, Linear
Algebra Appl., 525 (2017). In addition, we present a detailed study of how to
recover eigenvectors of a rational matrix from those of its linearizations in
this family. We complete the paper by discussing how to extend the results when
the polynomial part is expressed in other bases, and by presenting strong
linearizations that preserve the structure of symmetric or Hermitian rational
matrices. A conclusion of this work is that the combination of the results in
this paper with those in Amparan et al., MIMS EPrint 2016.51, allows us to use
essentially all the strong linearizations of polynomial matrices developed in
the last fifteen years to construct strong linearizations of any rational
matrix by expressing such matrix in terms of its polynomial and strictly proper
parts.
