Algebras behind the bispectrality of the Wilson rational functions and their ${}_4{\phi }_3$ limits

The properties of the Wilson rational functions ${}_{10}{\phi }_9$ with three different normalizations are described. For one normalization, it satisfies an $R_{II}$ recurrence relation, whereas for the two other ones, they satisfy a generalized eigenvalue problem. The so-called Wilson rational algebra is introduced, which encodes algebraically the spectral properties of these special functions. Finally, different limits are considered, leading up to functions proportional to ${}_4{\phi }_3$. For one of these, the spectral algebra simplifies to yield the meta $q$-Racah algebra.