Summary: We consider a singular differential-difference operator \(\Lambda\) on the real line which includes, as, particular case, the Dunkl operator associated with the reflection group \(\mathbb Z_2\) on \(\mathbb R\). We exhibit a Laplace integral representation for the eigenfunctions of the operator \(\Lambda\). From this representation, we construct a pair of integral transforms which turn out to be transmutation operators of \(\Lambda\) into the first derivative operator \(d/dx\). We exploit these transmutation operators to develop a new commutative harmonic analysis on the real line corresponding to the operator \(\Lambda\). In particular, we establish a Paley-Wiener theorem and a Plancherel theorem for the Fourier transform associated to \(\Lambda\).