  We introduce an algebra $\mathcal H$ consisting of difference-reflection
operators and multiplication operators that can be considered as a $q=1$
analogue of Sahi's double affine Hecke algebra related to the affine root
system of type $(C^\vee_1, C_1)$. We study eigenfunctions of a
Dunkl-Cherednik-type operator in the algebra $\mathcal H$, and the
corresponding Fourier transforms. These eigenfunctions are non-symmetric
versions of the Wilson polynomials and the Wilson functions.
