Abstract
We consider on ℝ the differential-difference operator
which is referred to as the Dunkl operator with parameter γ+1/2 associated with the reflection ℤ
2
on ℝ. We exhibit a Sonine-type integral representation for the Dunkl kernel e
γ
, eigenfunction of the Dunkl operator Λ
γ
. This enables us to construct a new Sonine integral transform on ℝ tied to Λ
γ
, which turns out together with its dual to be transmutation operators between two Dunkl operators of different parameters.