In this paper, we apply a limit as n→∞ from the little−1 Jacobi polynomials operator to obtain a new singular first-order differential-difference operator Y_α on the real line. The eigenfunction of this operator extends the Hartley kernel. Also, we give a special case of the Jordan algebra that has the operator Y_α as one of their three generators. Further, we study some elements of harmonic analysis related to Y_α.