Spectral transformation chains and some new biorthogonal rational functions (Q1976639)

Until recently Wilson's family of biorthogonal rational functions (very-well-poised, 2-balanced \({}_9F_8\) hypergeometric series with five free parameters) and their basic analogues due to Rahman and Wilson (very-well-poised, balanced basic \({}_{10}\varphi_9\) hypergeometric series with six free parameters) were considered the most general type of biorthogonal rational functions. The authors treat three even more general classes, starting from \({}_9F_8\) and \({}_{10}\varphi_9\) series as before, but now with a more intricate parametrization, and from elliptic analogues [due to \textit{I. B. Frenklin} and \textit{V. G. Turaev}, The Arnold-Gelfand Mathematical Seminars, Boston, MA, Birkhaüser, 171-204 (1997)] of the very-well-poised, balanced hypergeometric series in the context of elliptic solutions of the Yang-Baxter equations; this generalization leads to a ten parameter family. In the paper the authors use a method of proof based on an analysis of solutions of a chain of spectral transformations for a recurrence relation introduced by Ismail and Masson in connection to so called \(R_{II}\)-type continued fractions. This is a highly technical, densely written paper.