Contiguous relations, continued fractions and orthogonality: An \(_ 8\varphi_ 7\) model
In this extremely interesting paper the authors start with a contiguous relation for the very well posed \(_8 \varphi_7\) basic hypergeometric function \[ W : =_8 \varphi_7 \left( \begin{matrix} a,\;q \sqrt a,\;- q \sqrt a,\;b, \;c,\;d,\;e, \;f \\ \sqrt a,\;- \sqrt a,\;{aq \over b}, {aq \over c}, \dots, {aq \over f} \end{matrix} ;{a^2q^2 \over bcdef} \right). \] From this they obtain a three-term recurrence relation and thus a continued fraction expansion for the ratio of two ``consecutive \(Ws\)''. By solving the recurrence relation completely, they find the value of the continued fraction. For special values of the parameters the continued fraction terminates. For this situation the authors prove that there is an associated system of biorthogonal rational functions. This finite, discrete biorthogonality is explicitly given in the paper.
scientific article; zbMATH DE number 903570
scientific article