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arXiv · 2502.04275

Algebras behind the bispectrality of the Wilson rational functions and their ${}_4\phi_3$ limits

Nicolas Crampe·Satoshi Tsujimoto·Luc Vinet·Alexei Zhedanov
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Abstract

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.

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Nicolas Crampe, Satoshi Tsujimoto, Luc Vinet, Alexei Zhedanov. 2025-02-06. Algebras behind the bispectrality of the Wilson rational functions and their ${}_4\phi_3$ limits. https://arxiv.org/abs/2502.04275

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