Algebras behind the bispectrality of the Wilson rational functions and their ${}_4φ_3$ limits
Nicolas Crampe, Satoshi Tsujimoto, Luc Vinet, Alexei Zhedanov
TL;DR
This work addresses the algebraic origin of bispectrality for Wilson-type biorthogonal rational functions (BRF) and their ${}_{4}\phi_{3}$ limits by constructing the Wilson rational algebra from the tridiagonal recurrence data, and showing how limiting procedures yield a meta $q$-Racah algebra that governs the bispectral structure. It demonstrates two representations of the algebra via pairs of tridiagonal operators, clarifies how the ${}_{4}\phi_{3}$ limits recover $q$-Racah objects, and highlights the unifying meta-algebraic framework that extends the Askey scheme to BRFs. The results provide a coherent route to encode spectral information in algebraic terms and suggest future work on representations, $q=1$ limits, and spectral reconstruction from algebra. Overall, the paper advances a representation-theoretic synthesis of BRFs and their limits, linking Wilson-type functions to meta Racah algebras and enriching the algebraic foundation of the Askey scheme.
Abstract
The properties of the Wilson rational functions ${}_{10}φ_9$ 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 ${}_{4}φ_3$. For one of these, the spectral algebra simplifies to yield the meta $q$-Racah algebra.