We investigate the homogeneous symmetric Macdonald polynomials P λ (X; q, t) for the specialization t = q k . We show an identity relying the polynomials P λ (X; q, q k ) and P λ 1−q 1−q k X; q, q k . As a consequence, we describe an operator whose eigenvalues characterize the polynomials P λ (X; q, q k ).