Fourier transforms related to a root system of rank 1

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 $(C_1^{\vee },C_1)$. 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.