  This note consists of two parts. The first part (\S 1 and \S 2) is a partial
review of the works by van Meer and Stokman (2010), van Meer (2011) and Stokman
(2014) which established a bispectral analogue of the Cherednik correspondence
between quantum affine Knizhnik-Zamolodchikov equations and the eigenvalue
problems of Macdonald type. In this review we focus on the rank one cases,
i.e., on the reduced type $A_1$ and the non-reduced type $(C_1^\vee,C_1)$, to
which the associated Macdonald-Koornwinder polynomials are the Rogers
polynomials and the Askey-Wilson polynomials, respectively. We give detailed
computations and formulas that may be difficult to find in the literature. The
second part (\S 3) is a complement of the first part, and is also a
continuation of our previous study (Y.-Y., 2022) on the parameter
specialization of Macdonald-Koornwinder polynomials, where we found four types
of specialization of the type $(C_1^\vee,C_1)$ parameters (which could be
called the Askey-Wilson parameters) to recover the type $A_1$. In this note, we
show that among the four specializations there is only one which is compatible
with the bispectral correspondence discussed in the first part.
